reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;
reserve t for ExtReal;

theorem
  x < y & w < z implies x + w < y + z
proof
  assume that
A1: x < y and
A2: w < z;
  -infty <= w by XXREAL_0:5;
  then per cases by XXREAL_0:1;
  suppose
A3: w = -infty;
A4: y <> -infty & z <> -infty by A1,A2,XXREAL_0:5;
    x <> +infty by A1,XXREAL_0:3;
    then x + w = -infty by A3,Def2;
    hence thesis by A4,Lm9,XXREAL_0:6;
  end;
  suppose
A5: -infty < w;
  per cases;
  suppose
A6: y = +infty;
A7: z <= +infty by XXREAL_0:4;
    y + z = +infty by A5,A2,A6,Def2;
    hence thesis by A1,A2,A6,A7,Lm8,XXREAL_0:4;
  end;
  suppose
A8: y <> +infty;
    -infty <= x by XXREAL_0:5;
    then y in REAL by A1,A8,XXREAL_0:14;
    then
A9: y + w < y + z by A2,Th43;
    z <= +infty by XXREAL_0:4;
    then w in REAL by A5,A2,XXREAL_0:14;
    then x + w < y + w by A1,Th43;
    hence thesis by A9,XXREAL_0:2;
  end;
  end;
end;
