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
for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
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
for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
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
for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
let p be Polynomial of n,L; :: thesis: for i being Element of NAT st i <= card (Support p) holds
for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
let i be Element of NAT ; :: thesis: ( i <= card (Support p) implies for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L ) )
assume A1: i <= card (Support p) ; :: thesis: for b being bag of st b in Support (Low p,T,i) holds
( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
let b be bag of ; :: thesis: ( b in Support (Low p,T,i) implies ( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L ) )
assume A2: b in Support (Low p,T,i) ; :: thesis: ( (Low p,T,i) . b = p . b & (Up p,T,i) . b = 0. L )
set l = Lower_Support p,T,i;
A3: b in Lower_Support p,T,i by A1, A2, Lm3;
thus (Low p,T,i) . b = p . b by A2, Th16; :: thesis: (Up p,T,i) . b = 0. L
(Lower_Support p,T,i) /\ (Upper_Support p,T,i) = {} by A1, Th19;
then not b in Upper_Support p,T,i by A3, XBOOLE_0:def 4;
then A4: not b in Support (Up p,T,i) by A1, Lm3;
b is Element of Bags n by POLYNOM1:def 14;
hence (Up p,T,i) . b = 0. L by A4, POLYNOM1:def 9; :: thesis: verum