let A be set ; :: thesis: for f being Element of Funcs (A,REAL) holds (RealFuncExtMult A) . (1,f) = f
let f be Element of Funcs (A,REAL); :: thesis: (RealFuncExtMult A) . (1,f) = f

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

suppose A = {} ; :: thesis: (RealFuncExtMult A) . (1,f) = f

then A1: f = {} ;
thus (RealFuncExtMult A) . (1,f) = multreal [;] (1,f) by Def4
.= f by A1 ; :: thesis: verum

end;

suppose A <> {} ; :: thesis: (RealFuncExtMult A) . (1,f) = f

then reconsider A = A as non empty set ;
reconsider f = f as Element of Funcs (A,REAL) ;
reconsider g = (RealFuncExtMult A) . (1,f) as Element of Funcs (A,REAL) ;

now

let x be Element of A; :: thesis: g . x = f . x
thus g . x = 1 * (f . x) by Th15
.= f . x ; :: thesis: verum

end;

hence (RealFuncExtMult A) . (1,f) = f by FUNCT_2:63; :: thesis: verum

end;

end;