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 A: f = {} ;
thus (RealFuncExtMult A) . 1,f = multreal [;] 1,f by Def4
.= f by A ; :: 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_2:113; :: thesis: verum

end;

end;