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
A17: for h being Element of H holds T1 . h = the_left_translation_of (h,P) and
A18: 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 8 :: thesis: ^ = ^
thus T1 . x = the_left_translation_of (x,P) by A17
.= T2 . x by A18 ; :: thesis: verum