reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th53:
  A in B & (C c= A or C in A) implies A-^C in B-^C
proof
  assume that
A1: A in B and
A2: C c= A or C in A;
  A c= B by A1,ORDINAL1:def 2;
  then C c= B by A2,ORDINAL1:def 2;
  then
A3: B = C+^(B-^C) by Def5;
  C c= A by A2,ORDINAL1:def 2;
  then A = C+^(A-^C) by Def5;
  hence thesis by A1,A3,Th22;
end;
