let IT1, IT2 be TermOrder of n; :: thesis: ( ( for p, q being bag of n holds
( [p,q] in IT1 iff ( p = q or ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) ) ) ) & ( for p, q being bag of n holds
( [p,q] in IT2 iff ( p = q or ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) ) ) ) implies IT1 = IT2 )
assume that
A49: for p, q being bag of n holds
( [p,q] in IT1 iff ( p = q or ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) ) ) and
A50: for p, q being bag of n holds
( [p,q] in IT2 iff ( p = q or ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) ) ) ; :: thesis: IT1 = IT2

now :: thesis: for a, b being object holds
( ( [a,b] in IT1 implies [a,b] in IT2 ) & ( [a,b] in IT2 implies [a,b] in IT1 ) )

let a, b be object ; :: thesis: ( ( [a,b] in IT1 implies [a,b] in IT2 ) & ( [a,b] in IT2 implies [b₁,b₂] in IT1 ) )

hereby :: thesis: ( [a,b] in IT2 implies [b₁,b₂] in IT1 )

assume A51: [a,b] in IT1 ; :: thesis: [a,b] in IT2
then consider p, q being object such that
A52: [a,b] = [p,q] and
A53: p in Bags n and
A54: q in Bags n by RELSET_1:2;
reconsider p = p, q = q as bag of n by A53, A54;

per cases ( p = q or p <> q ) ;

suppose p = q ; :: thesis: [a,b] in IT2

hence [a,b] in IT2 by A50, A52; :: thesis: verum

end;

suppose p <> q ; :: thesis: [a,b] in IT2

then ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) by A49, A51, A52;
hence [a,b] in IT2 by A50, A52; :: thesis: verum

end;

end;

end;

assume A55: [a,b] in IT2 ; :: thesis: [b₁,b₂] in IT1
then consider p, q being object such that
A56: [a,b] = [p,q] and
A57: p in Bags n and
A58: q in Bags n by RELSET_1:2;
reconsider p = p, q = q as bag of n by A57, A58;

per cases ( p = q or p <> q ) ;

suppose p = q ; :: thesis: [b₁,b₂] in IT1

hence [a,b] in IT1 by A49, A56; :: thesis: verum

end;

suppose p <> q ; :: thesis: [b₁,b₂] in IT1

then ex i being Ordinal st
( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds
p . k = q . k ) ) by A50, A55, A56;
hence [a,b] in IT1 by A49, A56; :: thesis: verum

end;

end;

end;

hence IT1 = IT2 by RELAT_1:def 2; :: thesis: verum