reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;

theorem Th13:
  for f being Ordinal-Sequence holds
  f is non-decreasing implies f|a is non-decreasing
  proof let f be Ordinal-Sequence;
    assume
A1: for b,c st b in c & c in dom f holds f.b c= f.c;
    let b,c; assume
A2: b in c & c in dom(f|a); then
A3: c in dom f & c in a by RELAT_1:57; then
    (f|a).b = f.b & (f|a).c = f.c by A2,FUNCT_1:49,ORDINAL1:10;
    hence thesis by A1,A2,A3;
  end;
