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

thus (RealFuncAdd A) . (((RealFuncExtMult A) . (a,f)),((RealFuncExtMult A) . (a,g))) = (RealFuncAdd A) . (((RealFuncExtMult A) . (aa,f)),((RealFuncExtMult A) . (aa,g)))
.= {} by A1
.= multreal [;] (a,((RealFuncAdd A) . (f,g))) by A1
.= (RealFuncExtMult A) . (a,((RealFuncAdd A) . (f,g))) by Def3 ; :: thesis: verum

end;

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

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

now :: thesis: for x being Element of A holds ((RealFuncAdd A) . (((RealFuncExtMult A) . [aa,f]),((RealFuncExtMult A) . [aa,g]))) . x = ((RealFuncExtMult A) . [aa,((RealFuncAdd A) . (f,g))]) . x

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

end;

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

end;

end;