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

theorem Th4:
  h = (ComplexFuncExtMult(A)).[a,f] iff for x being Element of A
  holds h.x = a*(f.x)
proof
  thus h = (ComplexFuncExtMult(A)).[a,f] implies for x being Element of A
  holds h.x = a*(f.x) by Def3;
  reconsider a as Element of COMPLEX by XCMPLX_0:def 2;
  now
    assume
A1: for x being Element of A holds h.x = a*(f.x);
    for x being Element of A holds h.x = ((ComplexFuncExtMult(A)).[a,f]).x
    proof
      let x be Element of A;
      thus h.x = a*(f.x) by A1
        .= ((ComplexFuncExtMult(A)).[a,f]).x by Def3;
    end;
    hence h = (ComplexFuncExtMult(A)).[a,f] by FUNCT_2:63;
  end;
  hence thesis;
end;
