reserve a, b, c, d, e for Complex;

theorem
  ((d - c)/b) * a + c = (1 - a/b)*c + (a/b) * d
proof
  per cases;
  suppose
A1: b = 0;
A2: a/b = a*b" by XCMPLX_0:def 9
      .= a*0 by A1;
    thus ((d - c)/b) * a + c = ((d - c)*b") * a + c by XCMPLX_0:def 9
      .= ((d - c)*0) * a + c by A1
      .= (1 - (a/b))*c + (a/b) * d by A2;
  end;
  suppose
A3: b <> 0;
    ((d - c)/b) * a + c = ((d - c)/b) * a + c*1
      .= ((d - c)/b) * a + c*(b/b) by A3,Lm5
      .= ((d - c)/b) * a + c*b/b by Lm8
      .= (d - c) * a / b + c*b/b by Lm8
      .= ((d-c)*a + c*b)/ b by Th62
      .= ((b-a)*c + a * d) / b
      .= (b-a)*c/b + a*d/b by Th62
      .= (b-a)*c/b + (a/b) * d by Lm8
      .= ((b - a)/b)*c + (a/b) * d by Lm8
      .= (b/b - a/b)*c + (a/b) * d by Th120
      .= (1 - a/b)*c + (a/b) * d by A3,Lm5;
    hence thesis;
  end;
end;
