let T1, T2 be LeftOperation of H,(Left_Cosets P); :: thesis: ( ( for h being Element of H holds T1 . h = the_left_translation_of h,P ) & ( for h being Element of H holds T2 . h = the_left_translation_of h,P ) implies T1 = T2 )
assume that
A15: for h being Element of H holds T1 . h = the_left_translation_of h,P and
A16: for h being Element of H holds T2 . h = the_left_translation_of h,P ; :: thesis: T1 = T2
let x be Element of H; :: according to FUNCT_2:def 9 :: thesis: ^ = ^
thus T1 . x = the_left_translation_of x,P by A15
.= T2 . x by A16 ; :: thesis: verum