let A be non empty set ; :: thesis: for f being Element of Funcs A,COMPLEX
for a, b being Complex holds (ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])] = (ComplexFuncExtMult A) . [(a * b),f]
let f be Element of Funcs A,COMPLEX ; :: thesis: for a, b being Complex holds (ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])] = (ComplexFuncExtMult A) . [(a * b),f]
let a, b be Complex; :: thesis: (ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])] = (ComplexFuncExtMult A) . [(a * b),f]

now

let x be Element of A; :: thesis: ((ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])]) . x = ((ComplexFuncExtMult A) . [(a * b),f]) . x
thus ((ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])]) . x = a * (((ComplexFuncExtMult A) . [b,f]) . x) by Th6
.= a * (b * (f . x)) by Th6
.= (a * b) * (f . x)
.= ((ComplexFuncExtMult A) . [(a * b),f]) . x by Th6 ; :: thesis: verum

end;

hence (ComplexFuncExtMult A) . [a,((ComplexFuncExtMult A) . [b,f])] = (ComplexFuncExtMult A) . [(a * b),f] by FUNCT_2:113; :: thesis: verum