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)
reconsider aa = a, bb = b as Element of REAL by XREAL_0:def 1;

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

suppose A1: 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))) = (RealFuncAdd A) . (((RealFuncExtMult A) . (aa,f)),((RealFuncExtMult A) . (bb,f)))
.= {} by A1
.= multreal [;] ((a + b),f) by A1
.= (RealFuncExtMult A) . ((a + b),f) by Def3 ; :: 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 :: thesis: for x being Element of A holds ((RealFuncAdd A) . (((RealFuncExtMult A) . [aa,f]),((RealFuncExtMult A) . [bb,f]))) . x = ((RealFuncExtMult A) . [(aa + bb),f]) . x

let x be Element of A; :: thesis: ((RealFuncAdd A) . (((RealFuncExtMult A) . [aa,f]),((RealFuncExtMult A) . [bb,f]))) . x = ((RealFuncExtMult A) . [(aa + bb),f]) . x
thus ((RealFuncAdd A) . (((RealFuncExtMult A) . [aa,f]),((RealFuncExtMult A) . [bb,f]))) . x = (((RealFuncExtMult A) . [aa,f]) . x) + (((RealFuncExtMult A) . [bb,f]) . x) by Th1
.= (a * (f . x)) + (((RealFuncExtMult A) . [bb,f]) . x) by Th4
.= (a * (f . x)) + (b * (f . x)) by Th4
.= (a + b) * (f . x)
.= ((RealFuncExtMult A) . [(aa + bb),f]) . x by Th4 ; :: thesis: verum

end;

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

end;

end;