let A be set ; :: thesis: for f being Element of Funcs A,REAL holds (RealFuncAdd A) . (RealFuncZero A),f = f
let f be Element of Funcs A,REAL ; :: thesis: (RealFuncAdd A) . (RealFuncZero A),f = f

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

suppose A = {} ; :: thesis: (RealFuncAdd A) . (RealFuncZero A),f = f

then A1: f = {} ;
thus (RealFuncAdd A) . (RealFuncZero A),f = addreal .: (RealFuncZero A),f by Def2
.= f by A1 ; :: thesis: verum

end;

suppose A <> {} ; :: thesis: (RealFuncAdd A) . (RealFuncZero A),f = 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) . (RealFuncZero A),f) . x = f . x
thus ((RealFuncAdd A) . (RealFuncZero A),f) . x = ((RealFuncZero A) . x) + (f . x) by Th10
.= 0 + (f . x) by FUNCOP_1:13
.= f . x ; :: thesis: verum

end;

hence (RealFuncAdd A) . (RealFuncZero A),f = f by FUNCT_2:113; :: thesis: verum

end;

end;