let n be Ordinal; :: thesis: for T being connected TermOrder of n
for p being Polynomial of n,L
for i being Element of NAT st i <= card () holds
( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )

let T be connected TermOrder of n; :: thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st i <= card () holds
( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )

let L be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being Polynomial of n,L
for i being Element of NAT st i <= card () holds
( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )

let p be Polynomial of n,L; :: thesis: for i being Element of NAT st i <= card () holds
( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )

let i be Element of NAT ; :: thesis: ( i <= card () implies ( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) ) )
set u = Upper_Support (p,T,i);
set pu = p | (Upper_Support (p,T,i));
set l = Lower_Support (p,T,i);
set pl = p | (Lower_Support (p,T,i));
assume i <= card () ; :: thesis: ( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )
then A1: Upper_Support (p,T,i) c= Support p by Def2;
Support (p | (Upper_Support (p,T,i))) = () /\ (Upper_Support (p,T,i)) by Th16;
hence Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) by ; :: thesis: Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i)
Support (p | (Lower_Support (p,T,i))) = () /\ (Lower_Support (p,T,i)) by Th16;
hence Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) by ; :: thesis: verum