reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;
reserve e1,e2 for ExtReal;
reserve h,h1,h2 for PartFunc of REAL,REAL;

theorem
  for h being one-to-one PartFunc of REAL, REAL st h|[.p,g.] is
  increasing holds (h|[.p,g.])"|(h.:[.p,g.]) is increasing
proof
  let h be one-to-one PartFunc of REAL, REAL;
  assume that
A1: h|[.p,g.] is increasing and
A2: (h|[.p,g.])"|(h.:[.p,g.]) is not increasing;
  consider y1,y2 be Real such that
A3: y1 in h.:[.p,g.] /\ dom((h|[.p,g.])") and
A4: y2 in h.:[.p,g.] /\ dom((h|[.p,g.])") and
A5: y1<y2 and
A6: ((h|[.p,g.])").y1 >= ((h|[.p,g.])").y2 by A2,Th20;
  reconsider yy1 = y1, yy2 = y2 as Real;
  y1 in h.:[.p,g.] by A3,XBOOLE_0:def 4;
  then
A7: yy1 in rng (h|[.p,g.]) by RELAT_1:115;
  y2 in h.:[.p,g.] by A4,XBOOLE_0:def 4;
  then
A8: yy2 in rng (h|[.p,g.]) by RELAT_1:115;
A9: h|[.p,g.]|[.p,g.] is increasing by A1;
    per cases;
    suppose
      ((h|[.p,g.])").y1 = ((h|[.p,g.])").y2;
      then y1 =(h|[.p,g.]).(((h|[.p,g.])").y2) by A7,FUNCT_1:35
        .=y2 by A8,FUNCT_1:35;
      hence contradiction by A5;
    end;
    suppose
A10:  ((h|[.p,g.])").y1 <> ((h|[.p,g.])").y2;
A11:  dom (h|[.p,g.])=dom ((h|[.p,g.])|[.p,g.]) by RELAT_1:72
        .=[.p,g.]/\dom(h|[.p,g.]) by RELAT_1:61;
      ((h|[.p,g.])").yy2 in REAL & ((h|[.p,g.])").yy1 in REAL
       by XREAL_0:def 1;
      then
A12:  ((h|[.p,g.])").yy2 in dom (h|[.p,g.]) & ((h|[.p,g.])").yy1 in dom (h|
      [.p,g.]) by A7,A8,PARTFUN2:60;

      ((h|[.p,g.])").y2 < ((h|[.p,g.])").y1 by A6,A10,XXREAL_0:1;
      then (h|[.p,g.]).(((h|[.p,g.])").y2) < (h|[.p,g.]).(((h|[.p,g.])").y1)
      by A9,A12,A11,Th20;
      then y2 < (h|[.p,g.]).(((h|[.p,g.])").y1) by A8,FUNCT_1:35;
      hence contradiction by A5,A7,FUNCT_1:35;
    end;
end;
