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

theorem Th33:
  z is real implies (z + x) - (z + y) = x - y
proof
  assume
A1: z is real;
  per cases by A1,XXREAL_0:14;
  suppose
A2: x = -infty;
    per cases by XXREAL_0:14;
    suppose
A3:   y = -infty;
      (z + -infty) - (z + -infty) = -infty - (z + -infty) by A1,Def2
        .= -infty - -infty by A1,Def2;
      hence thesis by A2,A3;
    end;
    suppose
A4:   y = +infty;
      (z + -infty) - (z + +infty) = -infty - (z + +infty) by A1,Def2
        .= -infty - +infty by A1,Def2;
      hence thesis by A2,A4;
    end;
    suppose
A5:   y in REAL;
      then consider a,b being Complex such that
A6:   z = a & y = b and
      z+y = a + b by A1,Def2;
      reconsider a,b as Real by A1,A5,A6;
A7:   a+b in REAL by XREAL_0:def 1;
      (z + -infty) - (z + y) = -infty - (z + y) by A1,Def2
        .= -infty by A6,A7,Def2
        .= -infty - y by A5,Def2;
      hence thesis by A2;
    end;
  end;
  suppose
A8: y = +infty;
    per cases by XXREAL_0:14;
    suppose
A9:   x = -infty;
      (z + -infty) - (z + +infty) = z + -infty - +infty by A1,Def2
        .= -infty - +infty by A1,Def2;
      hence thesis by A8,A9;
    end;
    suppose
A10:  x = +infty;
      (z + +infty) - (z + +infty) = z + +infty - +infty by A1,Def2
        .= +infty - +infty by A1,Def2;
      hence thesis by A8,A10;
    end;
    suppose
A11:  x in REAL;
      then consider a,b being Complex such that
A12:  z = a & x = b and
      z+x = a + b by A1,Def2;
      reconsider a,b as Real by A1,A11,A12;
A13:  a+b in REAL by XREAL_0:def 1;
A14:  - +infty = -infty by Def3;
      (z + x) - (z + +infty) = z + x - +infty by A1,Def2
        .= z + x + -infty by Def3
        .= -infty by A12,A13,Def2
        .= x + - +infty by A11,A14,Def2;
      hence thesis by A8;
    end;
  end;
  suppose
    x in REAL & y in REAL & z in REAL;
    then reconsider a = x, b = y, c = z as Real;
    (z + x) - (z + y) = (a + c) - (c + b) .= a - b
      .= x - y;
    hence thesis;
  end;
  suppose
A15: x = +infty & y in REAL & z in REAL;
    then reconsider c = z, b = y as Real;
A16: z + y = c + b;
    z + x = +infty & x - y = +infty by A15,Def2;
    hence thesis by A16,Th13;
  end;
  suppose
A17: x in REAL & y = -infty & z in REAL;
    then reconsider c = z, a = x as Real;
    z + x = c + a & z + y = -infty by A17,Def2;
    then (z + x) - (z + y) = +infty by Th14
      .= x - y by A17,Th14;
    hence thesis;
  end;
  suppose
A18: x = +infty & y = -infty & z in REAL;
    then z + y = -infty & not z + x = -infty by Def2;
    then (z + x) - (z + y) = +infty by Th14
      .= x - y by A18,Th14;
    hence thesis;
  end;
end;
