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 Th51:
  R is well-ordering & S is well-ordering & a in field R & b in
field S & c in field S & R,S |_2 (S-Seg(b)) are_isomorphic & R |_2 (R-Seg(a)),S
  |_2 (S-Seg(c)) are_isomorphic implies S-Seg(c) c= S-Seg(b) & [c,b] in S
proof
  assume that
A1: R is well-ordering and
A2: S is well-ordering and
A3: a in field R and
A4: b in field S and
A5: c in field S and
A6: R,S |_2 (S-Seg(b)) are_isomorphic and
A7: R |_2 (R-Seg(a)),S |_2 (S-Seg(c)) are_isomorphic;
  set Q = S |_2 (S-Seg(b));
  set F1 = canonical_isomorphism_of(R,Q);
A8: F1 is_isomorphism_of R,Q by A1,A6,Def9;
  then consider d such that
A9: d in field Q and
A10: F1.:(R-Seg(a)) = Q-Seg(d) by A3,Th49;
A11: S-Seg(b) = field Q by A2,Th32;
  then
A12: Q-Seg(d) = S-Seg(d) by A2,A9,Th27;
A13: rng F1 = S-Seg(b) by A8,A11;
  then
A14: Q-Seg(d) c= S-Seg(b) by A10,RELAT_1:111;
  set T = S |_2 (S-Seg(c));
  set P = R |_2 (R-Seg(a));
A15: T,P are_isomorphic by A7,Th40;
A16: d in field S by A9,Th12;
  R-Seg(a) c= field R by Th9;
  then P,Q |_2 (F1.:(R-Seg(a))) are_isomorphic by A1,A8,Th48;
  then T,Q |_2 (Q-Seg(d)) are_isomorphic by A10,A15,Th42;
  then T,S |_2 (S-Seg(d)) are_isomorphic by A10,A12,A13,Th22,RELAT_1:111;
  hence S-Seg(c) c= S-Seg(b) by A2,A5,A12,A14,A16,Th47;
  hence thesis by A2,A4,A5,Th29;
end;
