reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem :: SQUARE_1'45
  max(a,b) = (a + b + |.a - b.|) / 2
proof
  per cases;
  suppose
A1: b <= a;
    hence max(a,b) = ((a+b)+ (a - b))/2 by XXREAL_0:def 10
      .= ((a+b)+|.a-b.|)/2 by A1,Th43,XREAL_1:48;
  end;
  suppose
A2: a <= b;
    then
A3: 0 <= b - a by XREAL_1:48;
    thus max(a,b) = ((a+b)+ -(a - b))/2 by A2,XXREAL_0:def 10
      .= ((a+b)+|.-(a-b).|)/2 by A3,Th43
      .= ((a+b)+|.a-b.|)/2 by Lm26;
  end;
end;
