let n be Ordinal; :: thesis: for T being TermOrder of n
for b1, b2 being bag of n holds
( max (b1,b2,T) = b1 or max (b1,b2,T) = b2 )
let T be TermOrder of n; :: thesis: for b1, b2 being bag of n holds
( max (b1,b2,T) = b1 or max (b1,b2,T) = b2 )
let b1, b2 be bag of n; :: thesis: ( max (b1,b2,T) = b1 or max (b1,b2,T) = b2 )
assume A1: max (b1,b2,T) <> b1 ; :: thesis: max (b1,b2,T) = b2

now :: thesis: ( ( not b2 <= b1,T & max (b1,b2,T) = b2 ) or ( b1 = b2 & contradiction ) )

per cases ( not b2 <= b1,T or b1 = b2 ) by A1, Def5;

case not b2 <= b1,T ; :: thesis: max (b1,b2,T) = b2

hence max (b1,b2,T) = b2 by Def5; :: thesis: verum

end;

case b1 = b2 ; :: thesis: contradiction

then b2 <= b1,T by Lm2;
hence contradiction by A1, Def5; :: thesis: verum

end;

end;

end;

hence max (b1,b2,T) = b2 ; :: thesis: verum