Then we must prove that lala lajpat rai was a British agent.
Not necessarily . By the time he suggested partition of Punjab and setting up other Muslim provinces in 1925 , the divide between Muslims and Hindus had already been created by the British .
To destroy the Hindu-Muslim unity the British relied upon their Muslim subjects. The Khilafat Movement was primarily a Muslim movement in which the Hindu participation was a matter of principle. The British encouraged the Muslims to break the unity fostered by Khilafat because they knew that this was the best manner of disillusioning the Hindus.
The first anti-Khilafat announcement was made by the Nizam of Hyderabad on 22 May 1920. His "firman" stated that since the Khilafat was an anti-Muslim movement it was henceforth declared illegal. During the Khilafat Movement Gandhi and the Ali Brothers had visited Aligarh Muslim University. To reduce the impact of their visit, the British incited the Muslims of Aligarh against the Hindus, on the pretext of protecting Muslim rights.
On the second of November 1921, Mohammad Shafi, a member of the Viceroy's Executive Council, presented a two
pronged scheme:
(i) To weaken the movement for a united India, itwas essential to lure away the Muslims; thiscould only be done if the British signed the Peace Treaty with Turkey.
(ii) An Anglo-Mohammadan Union in the interest of the British Empire should be organized.
On 21 September 1922, Viceroy Lord Reading wrote to
the Secretary of State for India:
I have just sent you a telegram, which will show you, how near we have been to a complete break between Muslims and Hindus. I have been giving the greatest attention to this possibility, and I have had the greatest assistance from Shafi on my council, who is a highly respectable Mohammadan.
As time passed the British realized that the success of their mission was dependant on the cooperation of the Muslims.
They decided that first they should announce their willingness to surrender control provided the Hindus and the Muslims were
ready to sink their differences. Then they should utilize their influence over the Muslims to make sure that the two parties
never cooperated. This was a sure way of creating dependence upon themselves. On this subject the Secretary of State for
India, Birkenhead wrote on 22 January 1925:
The more it is made obvious that these antagonisms are profound, and affect an immense and irreconcilable
section of the population, the more conspicuously is the fact illustrated that we and we alone can play the part
of the composers. 
Pakistan was given because League was creating lots of hurdles for peaceful transition of power like constituting the constituent assembly, princely states etc. Just after independence Liaquat Ali Khan even dissolved the constituent Assembly of Pakistan which delayed the framing of constitution in Pakistan. Whole world know who succeeded, who didn't.
