let IT1, IT2 be TermOrder of n; :: thesis: ( ( for p, q being bag of n holds
( [p,q] in IT1 iff ( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) ) ) & ( for p, q being bag of n holds
( [p,q] in IT2 iff ( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) ) ) implies IT1 = IT2 )
assume that
A44: for p, q being bag of n holds
( [p,q] in IT1 iff ( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) ) and
A45: for p, q being bag of n holds
( [p,q] in IT2 iff ( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) ) ; :: thesis: IT1 = IT2

now

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

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

assume A46: [a,b] in IT1 ; :: thesis: [a,b] in IT2
then A47: a in dom IT1 by RELAT_1:20;
b in rng IT1 by A46, RELAT_1:20;
then reconsider p = a, q = b as bag of n by A47;
( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) by A44, A46;
hence [a,b] in IT2 by A45; :: thesis: verum

end;

assume A48: [a,b] in IT2 ; :: thesis: [a,b] in IT1
then A49: a in dom IT2 by RELAT_1:20;
b in rng IT2 by A48, RELAT_1:20;
then reconsider p = a, q = b as bag of n by A49;
( ( 0 ,(i + 1) -cut p <> 0 ,(i + 1) -cut q & [(0 ,(i + 1) -cut p),(0 ,(i + 1) -cut q)] in o1 ) or ( 0 ,(i + 1) -cut p = 0 ,(i + 1) -cut q & [((i + 1),n -cut p),((i + 1),n -cut q)] in o2 ) ) by A45, A48;
hence [a,b] in IT1 by A44; :: thesis: verum

end;

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