let it1, it2 be Function of [:COMPLEX ,(Funcs A,COMPLEX ):],(Funcs A,COMPLEX ); :: thesis: ( ( for z being Complex
for f being Element of Funcs A,COMPLEX
for x being Element of A holds (it1 . [z,f]) . x = z * (f . x) ) & ( for z being Complex
for f being Element of Funcs A,COMPLEX
for x being Element of A holds (it2 . [z,f]) . x = z * (f . x) ) implies it1 = it2 )
assume that
A2: for z being Element of COMPLEX
for f being Element of Funcs A,COMPLEX
for x being Element of A holds (it1 . [z,f]) . x = z * (f . x) and
A3: for z being Element of COMPLEX
for f being Element of Funcs A,COMPLEX
for x being Element of A holds (it2 . [z,f]) . x = z * (f . x) ; :: thesis: it1 = it2

now

let z be Element of COMPLEX ; :: thesis: for f being Element of Funcs A,COMPLEX holds it1 . z,f = it2 . z,f
let f be Element of Funcs A,COMPLEX ; :: thesis: it1 . z,f = it2 . z,f

now

let x be Element of A; :: thesis: (it1 . [z,f]) . x = (it2 . [z,f]) . x
thus (it1 . [z,f]) . x = z * (f . x) by A2
.= (it2 . [z,f]) . x by A3 ; :: thesis: verum

end;

hence it1 . z,f = it2 . z,f by FUNCT_2:113; :: thesis: verum

end;

hence it1 = it2 by BINOP_1:2; :: thesis: verum