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

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

suppose A = {} ; :: thesis: (RealFuncMult A) . (RealFuncUnit A),f = f

then A1: f = {} ;
thus (RealFuncMult A) . (RealFuncUnit A),f = multreal .: (RealFuncUnit A),f by Def3
.= f by A1 ; :: thesis: verum

end;

suppose A <> {} ; :: thesis: (RealFuncMult A) . (RealFuncUnit 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: ((RealFuncMult A) . (RealFuncUnit A),f) . x = f . x
thus ((RealFuncMult A) . (RealFuncUnit A),f) . x = ((RealFuncUnit A) . x) * (f . x) by Th11
.= 1 * (f . x) by FUNCOP_1:13
.= f . x ; :: thesis: verum

end;

hence (RealFuncMult A) . (RealFuncUnit A),f = f by FUNCT_2:113; :: thesis: verum

end;

end;