let IT1, IT2 be TermOrder of n; :: thesis: ( ( for a, b being bag of holds
( [a,b] in IT1 iff ( TotDegree a < TotDegree b or ( TotDegree a = TotDegree b & [a,b] in o ) ) ) ) & ( for a, b being bag of holds
( [a,b] in IT2 iff ( TotDegree a < TotDegree b or ( TotDegree a = TotDegree b & [a,b] in o ) ) ) ) implies IT1 = IT2 )
assume that
A36: for a, b being bag of holds
( [a,b] in IT1 iff ( TotDegree a < TotDegree b or ( TotDegree a = TotDegree b & [a,b] in o ) ) ) and
A37: for a, b being bag of holds
( [a,b] in IT2 iff ( TotDegree a < TotDegree b or ( TotDegree a = TotDegree b & [a,b] in o ) ) ) ; :: 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 A38: [a,b] in IT1 ; :: thesis: [a,b] in IT2
then A39: a in dom IT1 by RELAT_1:def 4;
b in rng IT1 by A38, RELAT_1:def 5;
then reconsider a' = a, b' = b as bag of by A39, PRE_POLY:def 12;
( TotDegree a' < TotDegree b' or ( TotDegree a' = TotDegree b' & [a',b'] in o ) ) by A36, A38;
hence [a,b] in IT2 by A37; :: thesis: verum

end;

assume A40: [a,b] in IT2 ; :: thesis: [a,b] in IT1
then A41: a in dom IT2 by RELAT_1:def 4;
b in rng IT2 by A40, RELAT_1:def 5;
then reconsider a' = a, b' = b as bag of by A41, PRE_POLY:def 12;
( TotDegree a' < TotDegree b' or ( TotDegree a' = TotDegree b' & [a',b'] in o ) ) by A37, A40;
hence [a,b] in IT1 by A36; :: thesis: verum

end;

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