reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th30:
  R is well-ordering & a in field R & b in field R implies ( R-Seg
  (a) c= R-Seg(b) iff a = b or a in R-Seg(b) )
proof
  assume
A1: R is well-ordering & a in field R & b in field R;
  thus R-Seg(a) c= R-Seg(b) implies a = b or a in R-Seg(b)
  proof
    assume R-Seg(a) c= R-Seg(b);
    then [a,b] in R by A1,Th29;
    hence thesis by Th1;
  end;
  now
    assume a in R-Seg(b);
    then [a,b] in R by Th1;
    hence R-Seg(a) c= R-Seg(b) by A1,Th29;
  end;
  hence thesis;
end;
