let n be Ordinal; :: thesis: for T being connected TermOrder of n
for b1, b2 being bag of holds
( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )
let T be connected TermOrder of n; :: thesis: for b1, b2 being bag of holds
( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )
let b1, b2 be bag of ; :: thesis: ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )

per cases ( b1 <= b2,T or b2 <= b1,T ) by Lm5;

suppose A1: b1 <= b2,T ; :: thesis: ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )

then min b1,b2,T = b1 by Def4;
hence ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T ) by A1, Lm2; :: thesis: verum

end;

suppose A2: b2 <= b1,T ; :: thesis: ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )

now

per cases ( b1 = b2 or b1 <> b2 ) ;

case A3: b1 = b2 ; :: thesis: ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )

then min b1,b2,T = b1 by Lm6;
hence ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T ) by A3, Lm2; :: thesis: verum

end;

case b1 <> b2 ; :: thesis: ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T )

then b2 < b1,T by A2, Def3;
then not b1 <= b2,T by Th5;
then min b1,b2,T = b2 by Def4;
hence ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T ) by A2, Lm2; :: thesis: verum

end;

end;

end;

hence ( min b1,b2,T <= b1,T & min b1,b2,T <= b2,T ) ; :: thesis: verum

end;

end;