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 Def1
.= 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 :: thesis: for x being Element of A holds ((RealFuncAdd A) . ((RealFuncZero A),f)) . x = f . x

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 Th1
.= 0 + (f . x)
.= f . x ; :: thesis: verum

end;

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

end;

end;