reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  for f being one-to-one PartFunc of REAL,REAL st p<=g & [.p,g.] c= dom
f & (f|[.p,g.] is increasing or f|[.p,g.] is decreasing) holds (f|[.p,g.])"|(f
  .:[.p,g.]) is continuous
proof
  let f be one-to-one PartFunc of REAL,REAL;
  assume that
A1: p<=g and
A2: [.p,g.] c= dom f and
A3: f|[.p,g.] is increasing or f|[.p,g.] is decreasing;
  reconsider p, g as Real;
  now
    per cases by A3;
    suppose
A4:   f|[.p,g.] is increasing;
A5:   ((f|[.p,g.])").:(f.:[.p,g.]) = ((f|[.p,g.])").: (rng (f|[.p,g.])) by
RELAT_1:115
        .= ((f|[.p,g.])").:(dom ((f|[.p,g.])")) by FUNCT_1:33
        .= rng ((f|[.p,g.])") by RELAT_1:113
        .= dom (f|[.p,g.]) by FUNCT_1:33
        .= dom f /\ [.p,g.] by RELAT_1:61
        .= [.p,g.] by A2,XBOOLE_1:28;
      (f|[.p,g.])"|(f.:[.p,g.]) is increasing by A4,RFUNCT_2:51;
      hence thesis by A1,A5,Th46;
    end;
    suppose
A6:   f|[.p,g.] is decreasing;
A7:   ((f|[.p,g.])").:(f.:[.p,g.]) = ((f|[.p,g.])").: (rng (f|[.p,g.]))
      by RELAT_1:115
        .= ((f|[.p,g.])").:(dom ((f|[.p,g.])")) by FUNCT_1:33
        .= rng ((f|[.p,g.])") by RELAT_1:113
        .= dom (f|[.p,g.]) by FUNCT_1:33
        .= dom f /\ [.p,g.] by RELAT_1:61
        .= [.p,g.] by A2,XBOOLE_1:28;
      (f|[.p,g.])"|(f.:[.p,g.]) is decreasing by A6,RFUNCT_2:52;
      hence thesis by A1,A7,Th46;
    end;
  end;
  hence thesis;
end;
