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

theorem
  0 <= x & 0 <= z & z + x < y implies z <= y
proof
  assume that
A1: 0 <= x and
A2: 0 <= z and
A3: z + x < y;
  x in REAL
  proof
A4: x <> +infty
    proof
      assume x = +infty;
      then +infty < y by A2,A3,Def2;
      hence thesis by XXREAL_0:4;
    end;
    assume not x in REAL;
    hence thesis by A1,A4,XXREAL_0:10,Lm16;
  end;
  then
A5: (z + x) - x = z by Th22;
  y - 0 = y by Th15;
  hence thesis by A1,A3,A5,Th37;
end;
