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

theorem :: SQUARE_1'34
  min(a,b) = (a + b - |.a - b.|) / 2
proof
  per cases;
  suppose
A1: a <= b;
    |.a - b.| = |.-(b - a).| .= |.b - a.| by Lm26
      .= b - a by A1,Th43,XREAL_1:48;
    hence thesis by A1,XXREAL_0:def 9;
  end;
  suppose
A2: b <= a;
    hence min(a,b) = ((a+b)- (a - b))/2 by XXREAL_0:def 9
      .= ((a+b)-|.a-b.|)/2 by A2,Th43,XREAL_1:48;
  end;
end;
