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
  (id R)|R is increasing
proof
  now
    let r1,r2;
    assume that
A1: r1 in R /\ dom (id R) and
A2: r2 in R /\ dom (id R) and
A3: r1<r2;
    r1 in R by A1,XBOOLE_0:def 4;
    then
A4: (id R).r1 = r1 by FUNCT_1:18;
    r2 in R by A2,XBOOLE_0:def 4;
    hence (id R).r1 < (id R).r2 by A3,A4,FUNCT_1:18;
  end;
  hence thesis by Th20;
end;
