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 Th15:
  (ComplexFuncMult(A)).(f,(ComplexFuncAdd(A)).(g,h)) = (
  ComplexFuncAdd(A)).((ComplexFuncMult(A)).(f,g),(ComplexFuncMult(A)).(f,h))
proof
  now
    let x be Element of A;
    thus ((ComplexFuncMult(A)).(f,(ComplexFuncAdd(A)).(g,h))).x = f.x * (((
    ComplexFuncAdd(A)).(g,h)).x) by Th2
      .= f.x * (g.x + h.x) by Th1
      .= (f.x * g.x) + (f.x * h.x)
      .= ((ComplexFuncMult(A)).(f,g)).x + (f.x * h.x) by Th2
      .= ((ComplexFuncMult(A)).(f,g)).x + ((ComplexFuncMult(A)).(f,h)).x by Th2
      .= ((ComplexFuncAdd(A)). ((ComplexFuncMult(A)).(f,g),(ComplexFuncMult(
    A)).(f,h))).x by Th1;
  end;
  hence thesis by FUNCT_2:63;
end;
