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