let a, b be object ; :: thesis: for R being Relation st R is well-ordering holds
R -Seg a,R -Seg b are_c=-comparable
let R be Relation; :: thesis: ( R is well-ordering implies R -Seg a,R -Seg b are_c=-comparable )
assume A1: R is well-ordering ; :: thesis: R -Seg a,R -Seg b are_c=-comparable
A2: now :: thesis: ( a in field R & b in field R implies R -Seg a,R -Seg b are_c=-comparable )
assume A3: ( a in field R & b in field R ) ; :: thesis: R -Seg a,R -Seg b are_c=-comparable
now :: thesis: ( a <> b implies R -Seg a,R -Seg b are_c=-comparable )
assume a <> b ; :: thesis: R -Seg a,R -Seg b are_c=-comparable
A4: now :: thesis: ( [b,a] in R implies R -Seg b c= R -Seg a )
assume A5: [b,a] in R ; :: thesis: R -Seg b c= R -Seg a
now :: thesis: for c being object st c in R -Seg b holds
c in R -Seg a
let c be object ; :: thesis: ( c in R -Seg b implies c in R -Seg a )
assume A6: c in R -Seg b ; :: thesis: c in R -Seg a
then A7: [c,b] in R by Th1;
then A8: [c,a] in R by A1, A5, Lm2;
c <> b by A6, Th1;
then c <> a by A1, A5, A7, Lm3;
hence c in R -Seg a by A8, Th1; :: thesis: verum
end;
hence R -Seg b c= R -Seg a ; :: thesis: verum
end;
now :: thesis: ( [a,b] in R implies R -Seg a c= R -Seg b )
assume A9: [a,b] in R ; :: thesis: R -Seg a c= R -Seg b
now :: thesis: for c being object st c in R -Seg a holds
c in R -Seg b
let c be object ; :: thesis: ( c in R -Seg a implies c in R -Seg b )
assume A10: c in R -Seg a ; :: thesis: c in R -Seg b
then A11: [c,a] in R by Th1;
then A12: [c,b] in R by A1, A9, Lm2;
c <> a by A10, Th1;
then c <> b by A1, A9, A11, Lm3;
hence c in R -Seg b by A12, Th1; :: thesis: verum
end;
hence R -Seg a c= R -Seg b ; :: thesis: verum
end;
hence R -Seg a,R -Seg b are_c=-comparable by A1, A3, A4, Lm4; :: thesis: verum
end;
hence R -Seg a,R -Seg b are_c=-comparable ; :: thesis: verum
end;
now :: thesis: ( ( R -Seg a = {} or R -Seg b = {} ) implies R -Seg a,R -Seg b are_c=-comparable )
assume ( R -Seg a = {} or R -Seg b = {} ) ; :: thesis: R -Seg a,R -Seg b are_c=-comparable
then ( R -Seg a c= R -Seg b or R -Seg b c= R -Seg a ) ;
hence R -Seg a,R -Seg b are_c=-comparable ; :: thesis: verum
end;
hence R -Seg a,R -Seg b are_c=-comparable by A2, Th2; :: thesis: verum