reserve r,s for Real;
reserve x,y,z for Real;
reserve r,r1,r2 for Element of REAL+;

theorem
  for X,Y being Subset of REAL st for x,y st x in X & y in Y holds x <= y
  ex z st for x,y st x in X & y in Y holds x <= z & z <= y
proof
  let X,Y be Subset of REAL;
  assume
A1: for x,y st x in X & y in Y holds x <= y;
  per cases;
  suppose
A2: X = 0 or Y = 0;
    take 1;
    thus thesis by A2;
  end;
  suppose that
A3: X <> 0 and
A4: Y <> 0;
    consider x1 being Element of REAL such that
A5: x1 in X by A3,SUBSET_1:4;
A6: X c= REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_1:1;
A7: Y c= REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_1:1;
A8: ex x2 being Element of REAL st x2 in Y by A4,SUBSET_1:4;
    thus thesis
    proof
      per cases;
      suppose that
A9:     X misses REAL+ and
A10:    Y misses [:{0},REAL+:];
        take z = 0;
        let x,y such that
A11:    x in X and
A12:    y in Y;
        ( not z in [:{0},REAL+:])& not x in REAL+ by A9,A11,ARYTM_0:5
,ARYTM_2:20,XBOOLE_0:3;
        hence x <= z by XXREAL_0:def 5;
        Y c= REAL+ by A7,A10,XBOOLE_1:73;
        then reconsider y9 = y,o = 0 as Element of REAL+ by A12,ARYTM_2:20;
        o <=' y9 by ARYTM_1:6;
        hence thesis by Lm1;
      end;
      suppose
A13:    Y meets [:{0},REAL+:];
        {r: [0,r] in X} c= REAL+
        proof
          let u be object;
          assume u in {r: [0,r] in X};
          then ex r st u = r & [0,r] in X;
          hence thesis;
        end;
        then reconsider X9 = {r: [0,r] in X} as Subset of REAL+;
        {r: [0,r] in Y} c= REAL+
        proof
          let u be object;
          assume u in {r: [0,r] in Y};
          then ex r st u = r & [0,r] in Y;
          hence thesis;
        end;
        then reconsider Y9 = {r : [0,r] in Y} as Subset of REAL+;
        consider e being object such that
A14:    e in Y and
A15:    e in [:{0},REAL+:] by A13,XBOOLE_0:3;
        consider u,y1 being object such that
A16:    u in {0} and
A17:    y1 in REAL+ and
A18:    e = [u,y1] by A15,ZFMISC_1:84;
        reconsider y1 as Element of REAL+ by A17;
        e in REAL by A14;
        then
A19:    [0,y1] in REAL by A16,A18,TARSKI:def 1;
        then reconsider y0 = [0,y1] as Real;
A20:    y0 in Y by A14,A16,A18,TARSKI:def 1;
        then
A21:    y1 in Y9;
A22:    y0 in [:{0},REAL+:] by Lm4,ZFMISC_1:87;
A23:    X c= [:{0},REAL+:]
        proof
          let u be object;
          assume
A24:      u in X;
          then reconsider x = u as Real;
          now
            assume x in REAL+;
            then
A25:        not x in [:{0},REAL+:] by ARYTM_0:5,XBOOLE_0:3;
            x <= y0 & not y0 in REAL+ by A1,A22,A20,A24,ARYTM_0:5,XBOOLE_0:3;
            hence contradiction by A25,XXREAL_0:def 5;
          end;
          hence thesis by A6,A24,XBOOLE_0:def 3;
        end;
        then consider e,x3 being object such that
A26:    e in {0} and
A27:    x3 in REAL+ and
A28:    x1 = [e,x3] by A5,ZFMISC_1:84;
        reconsider x3 as Element of REAL+ by A27;
        x1 = [0,x3] by A26,A28,TARSKI:def 1;
        then
A29:    x3 in X9 by A5;
        for y9,x9 being Element of REAL+ st y9 in Y9 & x9 in X9 holds y9 <=' x9
        proof
          let y9,x9 be Element of REAL+;
          assume y9 in Y9;
          then
A30:      ex r1 st y9 = r1 & [0,r1] in Y;
          assume x9 in X9;
          then
A31:      ex r2 st x9 = r2 & [0,r2] in X;
          then reconsider x = [0,x9], y = [0,y9] as Real by A30;
A32:      x in [:{0},REAL+:] & y in [:{0},REAL+:] by Lm4,ZFMISC_1:87;
          x <= y by A1,A30,A31;
          then consider x99,y99 being Element of REAL+ such that
A33:      x = [0,x99] and
A34:      y = [0,y99] & y99 <=' x99 by A32,XXREAL_0:def 5;
          x9 = x99 by A33,XTUPLE_0:1;
          hence thesis by A34,XTUPLE_0:1;
        end;
        then consider z9 being Element of REAL+ such that
A35:    for y9,x9 being Element of REAL+ st y9 in Y9 & x9 in X9 holds
        y9 <=' z9 & z9 <=' x9 by A29,ARYTM_2:8;
A36:    y1 <> 0 by A19,ARYTM_0:3;
        y1 <=' z9 by A21,A29,A35;
        then [0,z9] in REAL by A36,ARYTM_0:2,ARYTM_1:5;
        then reconsider z = [0,z9] as Real;
        take z;
        let x,y such that
A37:    x in X and
A38:    y in Y;
        consider e,x9 being object such that
A39:    e in {0} and
A40:    x9 in REAL+ and
A41:    x = [e,x9] by A23,A37,ZFMISC_1:84;
        reconsider x9 as Element of REAL+ by A40;
A42:    z in [:{0},REAL+:] by Lm4,ZFMISC_1:87;
A43:    x = [0,x9] by A39,A41,TARSKI:def 1;
        then
A44:    x9 in X9 by A37;
        per cases by A7,A38,XBOOLE_0:def 3;
        suppose
A45:      y in REAL+;
          z9 <=' x9 by A21,A35,A44;
          hence x <= z by A23,A42,A37,A43,Lm1;
A46:      not y in [:{0},REAL+:] by A45,ARYTM_0:5,XBOOLE_0:3;
          not z in REAL+ by A42,ARYTM_0:5,XBOOLE_0:3;
          hence z <= y by A46,XXREAL_0:def 5;
        end;
        suppose
A47:      y in [:{0},REAL+:];
          then consider e,y9 being object such that
A48:      e in {0} and
A49:      y9 in REAL+ and
A50:      y = [e,y9] by ZFMISC_1:84;
          reconsider y9 as Element of REAL+ by A49;
A51:      y = [0,y9] by A48,A50,TARSKI:def 1;
          then y9 in Y9 by A38;
          then y9 <=' z9 & z9 <=' x9 by A35,A44;
          hence thesis by A23,A42,A37,A43,A47,A51,Lm1;
        end;
      end;
      suppose
A52:    X meets REAL+;
        reconsider X9 = X /\ REAL+ as Subset of REAL+ by XBOOLE_1:17;
        consider x1 being object such that
A53:    x1 in X and
A54:    x1 in REAL+ by A52,XBOOLE_0:3;
        reconsider x1 as Element of REAL+ by A54;
        x1 in REAL+;
        then reconsider x0 = x1 as Real by ARYTM_0:1;
A55:    Y c= REAL+
        proof
          let u be object;
          assume
A56:      u in Y;
          then reconsider y = u as Real;
          now
            assume y in [:{0},REAL+:];
            then
A57:        not y in REAL+ by ARYTM_0:5,XBOOLE_0:3;
            x0 <= y & not x0 in [:{0},REAL+:]
            by A1,A53,A56,ARYTM_0:5,XBOOLE_0:3;
            hence contradiction by A57,XXREAL_0:def 5;
          end;
          hence thesis by A7,A56,XBOOLE_0:def 3;
        end;
        then reconsider Y9 = Y as Subset of REAL+;
        for x9,y9 being Element of REAL+ st x9 in X9 & y9 in Y9 holds x9 <=' y9
        proof
          let x9,y9 be Element of REAL+;
A58:      X9 c= X by XBOOLE_1:17;
          x9 in REAL+ & y9 in REAL+;
          then reconsider x = x9, y = y9 as Real by ARYTM_0:1;
          assume x9 in X9 & y9 in Y9;
          then x <= y by A1,A58;
          then ex x9,y9 being Element of REAL+ st x = x9 & y = y9 & x9 <=' y9
          by XXREAL_0:def 5;
          hence thesis;
        end;
        then consider z9 being Element of REAL+ such that
A59:    for x9,y9 being Element of REAL+ st x9 in X9 & y9 in Y9
        holds x9 <=' z9 & z9 <=' y9 by A8,ARYTM_2:8;
        z9 in REAL+;
        then reconsider z = z9 as Real by ARYTM_0:1;
        take z;
        let x,y such that
A60:    x in X and
A61:    y in Y;
        reconsider y9 = y as Element of REAL+ by A55,A61;
A62:    x0 in X9 by A53,XBOOLE_0:def 4;
        per cases by A6,A60,XBOOLE_0:def 3;
        suppose x in REAL+;
          then reconsider x9 = x as Element of REAL+;
          x9 in X9 by A60,XBOOLE_0:def 4;
          then x9 <=' z9 & z9 <=' y9 by A59,A61;
          hence thesis by Lm1;
        end;
        suppose
A63:      x in [:{0},REAL+:];
A64:      not z in [:{0},REAL+:] by ARYTM_0:5,XBOOLE_0:3;
          not x in REAL+ by A63,ARYTM_0:5,XBOOLE_0:3;
          hence x <= z by A64,XXREAL_0:def 5;
          z9 <=' y9 by A62,A59,A61;
          hence z <= y by Lm1;
        end;
      end;
    end;
  end;
end;
