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_2:113; :: thesis: verum

end;

end;