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

theorem Th49:
  0 <= x & x < z implies ex y being Real st 0 < y & x + y < z
proof
  assume that
A1: 0 <= x and
A2: x < z;
  per cases by A1;
  suppose
A3: 0 < x;
    then 0 < z - x by A2,Th46;
    then consider y being Real such that
A4: 0 < y and
A5: y < z - x by Th3;
    take y;
A6: x + y <= x + (z - x) by A5,Th36;
A7: x in REAL by A2,A3,XXREAL_0:48;
    then
A8: x + (z - x) = z by Th22;
    x + y <> z by A7,A5,Th22;
    hence thesis by A4,A6,A8,XXREAL_0:1;
  end;
  suppose
A9: 0 = x;
    consider y being Real such that
A10: 0 < y & y < z by A1,A2,Th3;
    take y;
    thus thesis by A9,A10,Th4;
  end;
end;
