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

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

suppose S: A = {} ; :: thesis: (RealFuncAdd A) . ((RealFuncExtMult A) . a,f),((RealFuncExtMult A) . b,f) = (RealFuncExtMult A) . (a + b),f

thus (RealFuncAdd A) . ((RealFuncExtMult A) . a,f),((RealFuncExtMult A) . b,f) = {} by S
.= multreal [;] (a + b),f by S
.= (RealFuncExtMult A) . (a + b),f by Def4 ; :: thesis: verum

end;

suppose A <> {} ; :: thesis: (RealFuncAdd A) . ((RealFuncExtMult A) . a,f),((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: ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,f])) . x = ((RealFuncExtMult A) . [(a + b),f]) . x
thus ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,f])) . x = (((RealFuncExtMult A) . [a,f]) . x) + (((RealFuncExtMult A) . [b,f]) . x) by Th10
.= (a * (f . x)) + (((RealFuncExtMult A) . [b,f]) . x) by Th15
.= (a * (f . x)) + (b * (f . x)) by Th15
.= (a + b) * (f . x)
.= ((RealFuncExtMult A) . [(a + b),f]) . x by Th15 ; :: thesis: verum

end;

hence (RealFuncAdd A) . ((RealFuncExtMult A) . a,f),((RealFuncExtMult A) . b,f) = (RealFuncExtMult A) . (a + b),f by FUNCT_2:113; :: thesis: verum

end;

end;