reserve x,y,w,z for ExtReal,
  a for Real;

theorem
  x <> +infty & y <> +infty & not ( x = +infty & y = +infty or x =
  -infty & y = -infty ) implies min(x,y) = (x + y - |.x - y.|) / 2
proof
  assume that
A1: x <> +infty and
A2: y <> +infty and
A3: not ( x = +infty & y = +infty or x = -infty & y = -infty );
  per cases;
  suppose
A4: x = -infty;
    then x + y = -infty & x - y = -infty by A2,A3,XXREAL_3:14,def 2;
    then
A5: x + y - |.x - y.| = -infty by XXREAL_3:14;
A6: min(x,y) = -infty by A4,XXREAL_0:44;
    thus thesis by A6,A5,XXREAL_3:86;
  end;
  suppose
    x <> -infty;
    then reconsider a = x as Element of REAL by A1,XXREAL_0:14;
      per cases;
      suppose
A7:     y = -infty;
        then x + y = -infty & x - y = +infty by A1,A3,XXREAL_3:14,def 2;
        then
A8:     x + y - |.x - y.| = -infty by XXREAL_3:14;
A9:    min(x,y) = -infty by A7,XXREAL_0:44;
        thus thesis by A9,A8,XXREAL_3:86;
      end;
      suppose
        y <> -infty;
        then reconsider b = y as Element of REAL by A2,XXREAL_0:14;
        x - y = a - b by SUPINF_2:3;
        then x + y = a + b & |.x - y.| = |.a-b.| by SUPINF_2:1;
        then
A10:    x + y - |.x - y.| = a + b - |.a-b qua Complex.| by SUPINF_2:3;
        (x + y - |.x - y.|) / 2 = (a+b-|.a-b qua Complex.|)/2 by A10,Th2;
        hence thesis by COMPLEX1:73;
      end;
  end;
end;
