reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem Th4:
  h = (RealFuncExtMult A).[a,f] iff
  for x being Element of A holds h.x = a*(f.x)
proof
  reconsider aa=a as Element of REAL by XREAL_0:def 1;
  thus h = (RealFuncExtMult A).[a,f] implies for x being Element of A holds h.
  x = a*(f.x)
  proof
    assume
A1: h = (RealFuncExtMult A).[a,f];
    let x be Element of A;
    h = (RealFuncExtMult A).(a,f) by A1;
    hence h.x = (multreal[;](aa,f)).x by Def3
      .= multreal.(aa,f.x) by FUNCOP_1:53
      .= a*(f.x) by BINOP_2:def 11;
  end;
  now
    assume
A2: for x being Element of A holds h.x = a*(f.x);
    for x being Element of A holds h.x = ((RealFuncExtMult A).[aa,f]).x
    proof
      let x be Element of A;
A3:   multreal[;](a,f) = (RealFuncExtMult A).(a,f) by Def3;
      thus h.x = a*(f.x) by A2
        .= multreal.(a,f.x) by BINOP_2:def 11
        .= ((RealFuncExtMult A).[aa,f]).x by A3,FUNCOP_1:53;
    end;
    hence h = (RealFuncExtMult A).(a,f) by FUNCT_2:63;
  end;
  hence thesis;
end;
