reserve x for set;
reserve a,b,c,d for ExtReal;

theorem Th2:
  for a,b,c being ExtReal holds
  a <= b & b <= c implies a <= c
proof
  let a,b,c be ExtReal;
  assume that
A1: a <= b and
A2: b <= c;
  per cases;
  suppose that
A3: a in REAL+ and
A4: b in REAL+ and
A5: c in REAL+;
    consider b99,c9 being Element of REAL+ such that
A6: b = b99 and
A7: c = c9 and
A8: b99 <=' c9 by A2,A4,A5,Def5;
    consider a9,b9 being Element of REAL+ such that
A9: a = a9 and
A10: b = b9 & a9 <=' b9 by A1,A3,A4,Def5;
    a9 <=' c9 by A10,A6,A8,ARYTM_1:3;
    hence thesis by A5,A9,A7,Def5;
  end;
  suppose
A11: a in REAL+ & b in [:{0},REAL+:];
    then
    ( not(a in REAL+ & b in REAL+))& not(a in [:{0},REAL+:] & b in [:{0},
    REAL+:]) by ARYTM_0:5,XBOOLE_0:3;
    hence thesis by A1,A11,Def5,Lm4,Lm5;
  end;
  suppose
A12: b in REAL+ & c in [:{0},REAL+:];
    then
    ( not(c in REAL+ & b in REAL+))& not(c in [:{0},REAL+:] & b in [:{0},
    REAL+:]) by ARYTM_0:5,XBOOLE_0:3;
    hence thesis by A2,A12,Def5,Lm4,Lm5;
  end;
  suppose that
A13: a in [:{0},REAL+:] & c in REAL+;
    ( not(a in REAL+ & c in REAL+))& not(a in [:{0},REAL+:] & c in [:{0},
    REAL+:]) by A13,ARYTM_0:5,XBOOLE_0:3;
    hence thesis by A13,Def5;
  end;
  suppose that
A14: a in [:{0},REAL+:] and
A15: b in [:{0},REAL+:] and
A16: c in [:{0},REAL+:];
    consider b99,c9 being Element of REAL+ such that
A17: b = [0,b99] and
A18: c = [0,c9] and
A19: c9 <=' b99 by A2,A15,A16,Def5;
    consider a9,b9 being Element of REAL+ such that
A20: a = [0,a9] and
A21: b = [0,b9] and
A22: b9 <=' a9 by A1,A14,A15,Def5;
    b9 = b99 by A21,A17,XTUPLE_0:1;
    then c9 <=' a9 by A22,A19,ARYTM_1:3;
    hence thesis by A14,A16,A20,A18,Def5;
  end;
  suppose that
A23: not(a in REAL+ & b in REAL+ & c in REAL+) and
A24: not(a in REAL+ & b in [:{0},REAL+:]) and
A25: not(b in REAL+ & c in [:{0},REAL+:]) and
A26: not(a in [:{0},REAL+:] & c in REAL+) and
A27: not(a in [:{0},REAL+:] & b in [:{0},REAL+:] & c in [:{0},REAL+:]);
A28: b = +infty implies c = +infty by A2,Lm9;
A29: b = -infty implies a = -infty by A1,Lm8;
    a = -infty or b = +infty or b = -infty or c = +infty by A1,A2,A23,A25,A26
,A27,Def5;
    hence thesis by A1,A2,A23,A24,A25,A27,A28,A29,Def5;
  end;
end;
