let b, a be set ; :: thesis: for R being Relation st R is well-ordering & b in R -Seg a holds
(R |_2 (R -Seg a)) -Seg b = R -Seg b
let R be Relation; :: thesis: ( R is well-ordering & b in R -Seg a implies (R |_2 (R -Seg a)) -Seg b = R -Seg b )
assume that
A1: R is well-ordering and
A2: b in R -Seg a ; :: thesis: (R |_2 (R -Seg a)) -Seg b = R -Seg b
set S = R |_2 (R -Seg a);
A3: ( R is transitive & R is reflexive & R is antisymmetric ) by A1, Def4;

now

let c be set ; :: thesis: ( c in R -Seg b implies c in (R |_2 (R -Seg a)) -Seg b )
assume A4: c in R -Seg b ; :: thesis: c in (R |_2 (R -Seg a)) -Seg b
then A5: [c,b] in R by Th1;
A6: [b,a] in R by A2, Th1;
then A7: [c,a] in R by A3, A5, Lm2;
A8: c <> b by A4, Th1;
then c <> a by A3, A5, A6, Lm3;
then c in R -Seg a by A7, Th1;
then [c,b] in [:(R -Seg a),(R -Seg a):] by A2, ZFMISC_1:87;
then [c,b] in R |_2 (R -Seg a) by A5, XBOOLE_0:def 4;
hence c in (R |_2 (R -Seg a)) -Seg b by A8, Th1; :: thesis: verum

end;

then A9: R -Seg b c= (R |_2 (R -Seg a)) -Seg b by TARSKI:def 3;

now

let c be set ; :: thesis: ( c in (R |_2 (R -Seg a)) -Seg b implies c in R -Seg b )
assume A10: c in (R |_2 (R -Seg a)) -Seg b ; :: thesis: c in R -Seg b
then [c,b] in R |_2 (R -Seg a) by Th1;
then A11: [c,b] in R by XBOOLE_0:def 4;
c <> b by A10, Th1;
hence c in R -Seg b by A11, Th1; :: thesis: verum

end;

then (R |_2 (R -Seg a)) -Seg b c= R -Seg b by TARSKI:def 3;
hence (R |_2 (R -Seg a)) -Seg b = R -Seg b by A9, XBOOLE_0:def 10; :: thesis: verum