let a, b be object ; :: thesis: for R being Relation st R is well-ordering & a in field R & b in field R holds
( [a,b] in R iff R -Seg a c= R -Seg b )
let R be Relation; :: thesis: ( R is well-ordering & a in field R & b in field R implies ( [a,b] in R iff R -Seg a c= R -Seg b ) )
assume that
A1: R is well-ordering and
A2: a in field R and
A3: b in field R ; :: thesis: ( [a,b] in R iff R -Seg a c= R -Seg b )
thus ( [a,b] in R implies R -Seg a c= R -Seg b ) :: thesis: ( R -Seg a c= R -Seg b implies [a,b] in R )

proof

assume A4: [a,b] in R ; :: thesis: R -Seg a c= R -Seg b
let c be object ; :: according to TARSKI:def 3 :: thesis: ( not c in R -Seg a or c in R -Seg b )
assume A5: c in R -Seg a ; :: thesis: c in R -Seg b
then A6: [c,a] in R by Th1;
then A7: [c,b] in R by A1, A4, Lm2;
c <> a by A5, Th1;
then c <> b by A1, A4, A6, Lm3;
hence c in R -Seg b by A7, Th1; :: thesis: verum

end;

assume A8: R -Seg a c= R -Seg b ; :: thesis: [a,b] in R

now :: thesis: ( a <> b implies [a,b] in R )

assume A9: a <> b ; :: thesis: [a,b] in R
assume not [a,b] in R ; :: thesis: contradiction
then [b,a] in R by A2, A3, A1, A9, Lm4;
then b in R -Seg a by A9, Th1;
hence contradiction by A8, Th1; :: thesis: verum

end;

hence [a,b] in R by A1, A2, Lm1; :: thesis: verum