reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th10:
  fi is increasing & A in dom fi implies A c= fi.A
proof
  assume that
A1: for A,B st A in B & B in dom fi holds fi.A in fi.B and
A2: A in dom fi and
A3: not A c= fi.A;
  defpred P[set] means $1 in dom fi & not $1 c= fi.$1;
A4: ex A st P[A] by A2,A3;
  consider A such that
A5: P[A] and
A6: for B st P[B] holds A c= B from ORDINAL1:sch 1(A4);
  reconsider B = fi.A as Ordinal;
A7: B in A by A5,ORDINAL1:16;
  then not B c= fi.B by A1,A5,ORDINAL1:5;
  hence contradiction by A5,A6,A7,ORDINAL1:10;
end;
