let n be Ordinal; :: thesis: for T being connected TermOrder of n
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 (Support p) holds
( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p )
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 (Support p) holds
( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p )
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 (Support p) holds
( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p )
let p be Polynomial of n,L; :: thesis: for i being Element of NAT st i <= card (Support p) holds
( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p )
let i be Element of NAT ; :: thesis: ( i <= card (Support p) implies ( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p ) )
assume A1: i <= card (Support p) ; :: thesis: ( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p )
then ( Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) ) by Lm3;
hence ( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p ) by A1, Def2, Th24; :: thesis: verum