let a, b be Real; :: thesis: for A being set
for f being Element of Funcs (A,REAL) holds (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)
let A be set ; :: thesis: for f being Element of Funcs (A,REAL) holds (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)
let f be Element of Funcs (A,REAL); :: thesis: (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)

per cases ( A = {} or A <> {} ) ;

suppose A1: A = {} ; :: thesis: (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)

(RealFuncExtMult A) . (b,f) = multreal [;] (b,f) by Def4;
hence (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = multreal [;] (a,(multreal [;] (b,f))) by Def4
.= multreal [;] ((a * b),f) by A1
.= (RealFuncExtMult A) . ((a * b),f) by Def4 ;
:: thesis: verum

end;

suppose A <> {} ; :: thesis: (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f)

then reconsider A = A as non empty set ;
reconsider f = f as Element of Funcs (A,REAL) ;

now

let x be Element of A; :: thesis: ((RealFuncExtMult A) . [a,((RealFuncExtMult A) . [b,f])]) . x = ((RealFuncExtMult A) . [(a * b),f]) . x
thus ((RealFuncExtMult A) . [a,((RealFuncExtMult A) . [b,f])]) . x = a * (((RealFuncExtMult A) . [b,f]) . x) by Th15
.= a * (b * (f . x)) by Th15
.= (a * b) * (f . x)
.= ((RealFuncExtMult A) . [(a * b),f]) . x by Th15 ; :: thesis: verum

end;

hence (RealFuncExtMult A) . (a,((RealFuncExtMult A) . (b,f))) = (RealFuncExtMult A) . ((a * b),f) by FUNCT_2:63; :: thesis: verum

end;

end;