:: GROEB_3 semantic presentation

begin

theorem Th1: :: GROEB_3:1
for X being set
for b1, b2 being bag of X holds (b1 + b2) / b2 = b1
proof
let X be set ; ::_thesis: for b1, b2 being bag of X holds (b1 + b2) / b2 = b1
let b1, b2 be bag of X; ::_thesis: (b1 + b2) / b2 = b1
 b2 divides b1 + b2 by PRE_POLY:50;
then  b2 + ((b1 + b2) / b2) = b1 + b2 by GROEB_2:def_1;
then  (b2 + ((b1 + b2) / b2)) -' b2 = b1 by PRE_POLY:48;
hence  (b1 + b2) / b2 = b1 by PRE_POLY:48; ::_thesis: verum
end;

theorem Th2: :: GROEB_3:2
for n being Ordinal
for T being admissible TermOrder of n
for b1, b2, b3 being bag of n st b1 <= b2,T holds
b1 + b3 <= b2 + b3,T
proof
let n be Ordinal; ::_thesis: for T being admissible TermOrder of n
for b1, b2, b3 being bag of n st b1 <= b2,T holds
b1 + b3 <= b2 + b3,T
let T be admissible TermOrder of n; ::_thesis: for b1, b2, b3 being bag of n st b1 <= b2,T holds
b1 + b3 <= b2 + b3,T
let b1, b2, b3 be bag of n; ::_thesis: ( b1 <= b2,T implies b1 + b3 <= b2 + b3,T )
assume  b1 <= b2,T ; ::_thesis: b1 + b3 <= b2 + b3,T
then  [b1,b2] in T by TERMORD:def_2;
then  [(b1 + b3),(b2 + b3)] in T by BAGORDER:def_5;
hence  b1 + b3 <= b2 + b3,T by TERMORD:def_2; ::_thesis: verum
end;

theorem Th3: :: GROEB_3:3
for n being Ordinal
for T being TermOrder of n
for b1, b2, b3 being bag of n st b1 <= b2,T & b2 < b3,T holds
b1 < b3,T
proof
let n be Ordinal; ::_thesis: for T being TermOrder of n
for b1, b2, b3 being bag of n st b1 <= b2,T & b2 < b3,T holds
b1 < b3,T
let T be TermOrder of n; ::_thesis: for b1, b2, b3 being bag of n st b1 <= b2,T & b2 < b3,T holds
b1 < b3,T
let b1, b2, b3 be bag of n; ::_thesis: ( b1 <= b2,T & b2 < b3,T implies b1 < b3,T )
assume that
A1: b1 <= b2,T and
A2: b2 < b3,T ; ::_thesis: b1 < b3,T
A3: b2 <= b3,T by A2, TERMORD:def_3;
then A4: b1 <= b3,T by A1, TERMORD:8;
 b2 <> b3 by A2, TERMORD:def_3;
then  b1 <> b3 by A1, A3, TERMORD:7;
hence  b1 < b3,T by A4, TERMORD:def_3; ::_thesis: verum
end;

theorem Th4: :: GROEB_3:4
for n being Ordinal
for T being admissible TermOrder of n
for b1, b2, b3 being bag of n st b1 < b2,T holds
b1 + b3 < b2 + b3,T
proof
let n be Ordinal; ::_thesis: for T being admissible TermOrder of n
for b1, b2, b3 being bag of n st b1 < b2,T holds
b1 + b3 < b2 + b3,T
let T be admissible TermOrder of n; ::_thesis: for b1, b2, b3 being bag of n st b1 < b2,T holds
b1 + b3 < b2 + b3,T
let b1, b2, b3 be bag of n; ::_thesis: ( b1 < b2,T implies b1 + b3 < b2 + b3,T )
assume A1: b1 < b2,T ; ::_thesis: b1 + b3 < b2 + b3,T
A2: now__::_thesis:_not_b1_+_b3_=_b2_+_b3
assume A3: b1 + b3 = b2 + b3 ; ::_thesis: contradiction
b1 = (b1 + b3) -' b3 by PRE_POLY:48
.= b2 by A3, PRE_POLY:48 ;
hence  contradiction by A1, TERMORD:def_3; ::_thesis: verum
end;
 b1 <= b2,T by A1, TERMORD:def_3;
then  [b1,b2] in T by TERMORD:def_2;
then  [(b1 + b3),(b2 + b3)] in T by BAGORDER:def_5;
then  b1 + b3 <= b2 + b3,T by TERMORD:def_2;
hence  b1 + b3 < b2 + b3,T by A2, TERMORD:def_3; ::_thesis: verum
end;

theorem Th5: :: GROEB_3:5
for n being Ordinal
for T being admissible TermOrder of n
for b1, b2, b3, b4 being bag of n st b1 < b2,T & b3 <= b4,T holds
b1 + b3 < b2 + b4,T
proof
let n be Ordinal; ::_thesis: for T being admissible TermOrder of n
for b1, b2, b3, b4 being bag of n st b1 < b2,T & b3 <= b4,T holds
b1 + b3 < b2 + b4,T
let T be admissible TermOrder of n; ::_thesis: for b1, b2, b3, b4 being bag of n st b1 < b2,T & b3 <= b4,T holds
b1 + b3 < b2 + b4,T
let b1, b2, b3, b4 be bag of n; ::_thesis: ( b1 < b2,T & b3 <= b4,T implies b1 + b3 < b2 + b4,T )
assume that
A1: b1 < b2,T and
A2: b3 <= b4,T ; ::_thesis: b1 + b3 < b2 + b4,T
 b1 <= b2,T by A1, TERMORD:def_3;
then  [b1,b2] in T by TERMORD:def_2;
then  [(b1 + b3),(b2 + b3)] in T by BAGORDER:def_5;
then A3: b1 + b3 <= b2 + b3,T by TERMORD:def_2;
 [b3,b4] in T by A2, TERMORD:def_2;
then  [(b2 + b3),(b2 + b4)] in T by BAGORDER:def_5;
then A4: b2 + b3 <= b2 + b4,T by TERMORD:def_2;
A5: now__::_thesis:_not_b1_+_b3_=_b2_+_b4
A6: ( b1 = (b1 + b4) -' b4 & b2 = (b2 + b4) -' b4 ) by PRE_POLY:48;
A7: ( b4 = (b4 + b2) -' b2 & b3 = (b3 + b2) -' b2 ) by PRE_POLY:48;
assume  b1 + b3 = b2 + b4 ; ::_thesis: contradiction
then  b1 + b4 = b2 + b4 by A3, A4, A7, TERMORD:7;
hence  contradiction by A1, A6, TERMORD:def_3; ::_thesis: verum
end;
 b1 + b3 <= b2 + b4,T by A3, A4, TERMORD:8;
hence  b1 + b3 < b2 + b4,T by A5, TERMORD:def_3; ::_thesis: verum
end;

theorem Th6: :: GROEB_3:6
for n being Ordinal
for T being admissible TermOrder of n
for b1, b2, b3, b4 being bag of n st b1 <= b2,T & b3 < b4,T holds
b1 + b3 < b2 + b4,T
proof
let n be Ordinal; ::_thesis: for T being admissible TermOrder of n
for b1, b2, b3, b4 being bag of n st b1 <= b2,T & b3 < b4,T holds
b1 + b3 < b2 + b4,T
let T be admissible TermOrder of n; ::_thesis: for b1, b2, b3, b4 being bag of n st b1 <= b2,T & b3 < b4,T holds
b1 + b3 < b2 + b4,T
let b1, b2, b3, b4 be bag of n; ::_thesis: ( b1 <= b2,T & b3 < b4,T implies b1 + b3 < b2 + b4,T )
assume that
A1: b1 <= b2,T and
A2: b3 < b4,T ; ::_thesis: b1 + b3 < b2 + b4,T
 b3 <= b4,T by A2, TERMORD:def_3;
then  [b3,b4] in T by TERMORD:def_2;
then  [(b2 + b3),(b2 + b4)] in T by BAGORDER:def_5;
then A3: b2 + b3 <= b2 + b4,T by TERMORD:def_2;
 [b1,b2] in T by A1, TERMORD:def_2;
then  [(b1 + b3),(b2 + b3)] in T by BAGORDER:def_5;
then A4: b1 + b3 <= b2 + b3,T by TERMORD:def_2;
A5: now__::_thesis:_not_b1_+_b3_=_b2_+_b4
assume  b1 + b3 = b2 + b4 ; ::_thesis: contradiction
then A6: b2 + b4 = b2 + b3 by A4, A3, TERMORD:7;
 ( b4 = (b4 + b2) -' b2 & b3 = (b3 + b2) -' b2 ) by PRE_POLY:48;
hence  contradiction by A2, A6, TERMORD:def_3; ::_thesis: verum
end;
 b1 + b3 <= b2 + b4,T by A4, A3, TERMORD:8;
hence  b1 + b3 < b2 + b4,T by A5, TERMORD:def_3; ::_thesis: verum
end;

begin

theorem Th7: :: GROEB_3:7
for n being Ordinal
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for m1, m2 being non-zero Monomial of n,L holds term (m1 *' m2) = (term m1) + (term m2)
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for m1, m2 being non-zero Monomial of n,L holds term (m1 *' m2) = (term m1) + (term m2)
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for m1, m2 being non-zero Monomial of n,L holds term (m1 *' m2) = (term m1) + (term m2)
let m1, m2 be non-zero Monomial of n,L; ::_thesis: term (m1 *' m2) = (term m1) + (term m2)
set T = the connected TermOrder of n;
A1: HC (m2, the connected TermOrder of n) <> 0. L ;
 HC (m1, the connected TermOrder of n) <> 0. L ;
then reconsider a = coefficient m1, b = coefficient m2 as non zero Element of L by A1, TERMORD:23;
 a * b <> 0. L by VECTSP_2:def_1;
then reconsider c = a * b as non zero Element of L by STRUCT_0:def_12;
 ( m1 = Monom (a,(term m1)) & m2 = Monom (b,(term m2)) ) by POLYNOM7:11;
then  term (m1 *' m2) = term (Monom (c,((term m1) + (term m2)))) by TERMORD:3;
hence  term (m1 *' m2) = (term m1) + (term m2) by POLYNOM7:10; ::_thesis: verum
end;

theorem Th8: :: GROEB_3:8
for n being Ordinal
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for b being bag of n holds
( b in Support p iff (term m) + b in Support (m *' p) )
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for b being bag of n holds
( b in Support p iff (term m) + b in Support (m *' p) )
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for b being bag of n holds
( b in Support p iff (term m) + b in Support (m *' p) )
let p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L
for b being bag of n holds
( b in Support p iff (term m) + b in Support (m *' p) )
let m be non-zero Monomial of n,L; ::_thesis: for b being bag of n holds
( b in Support p iff (term m) + b in Support (m *' p) )
let b be bag of n; ::_thesis: ( b in Support p iff (term m) + b in Support (m *' p) )
A1: (m *' p) . ((term m) + b) = (m . (term m)) * (p . b) by POLYRED:7;
 m <> 0_ (n,L) by POLYNOM7:def_1;
then  Support m <> {} by POLYNOM7:1;
then  Support m = {(term m)} by POLYNOM7:7;
then  term m in Support m by TARSKI:def_1;
then A2: m . (term m) <> 0. L by POLYNOM1:def_3;
A3: now__::_thesis:_(_b_in_Support_p_implies_(term_m)_+_b_in_Support_(m_*'_p)_)
assume  b in Support p ; ::_thesis: (term m) + b in Support (m *' p)
then  p . b <> 0. L by POLYNOM1:def_3;
then  ( (term m) + b is Element of Bags n & (m *' p) . ((term m) + b) <> 0. L ) by A2, A1, PRE_POLY:def_12, VECTSP_2:def_1;
hence  (term m) + b in Support (m *' p) by POLYNOM1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_(term_m)_+_b_in_Support_(m_*'_p)_implies_b_in_Support_p_)
assume  (term m) + b in Support (m *' p) ; ::_thesis: b in Support p
then  (m . (term m)) * (p . b) <> 0. L by A1, POLYNOM1:def_3;
then A4: p . b <> 0. L by VECTSP_1:6;
 b is Element of Bags n by PRE_POLY:def_12;
hence  b in Support p by A4, POLYNOM1:def_3; ::_thesis: verum
end;
hence  ( b in Support p iff (term m) + b in Support (m *' p) ) by A3; ::_thesis: verum
end;

theorem Th9: :: GROEB_3:9
for n being Ordinal
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
let p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L holds Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
let m be non-zero Monomial of n,L; ::_thesis: Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
 m <> 0_ (n,L) by POLYNOM7:def_1;
then  Support m <> {} by POLYNOM7:1;
then A1: Support m = {(term m)} by POLYNOM7:7;
A2: Support (m *' p) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by TERMORD:30;
A3: now__::_thesis:_for_u_being_set_st_u_in_Support_(m_*'_p)_holds_
u_in__{__((term_m)_+_b)_where_b_is_Element_of_Bags_n_:_b_in_Support_p__}_
let u be set ; ::_thesis: ( u in Support (m *' p) implies u in  {  ((term m) + b) where b is Element of Bags n : b in Support p  }  )
assume A4: u in Support (m *' p) ; ::_thesis: u in  {  ((term m) + b) where b is Element of Bags n : b in Support p  }
then reconsider u9 = u as Element of Bags n ;
 u9 in  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by A2, A4;
then consider s, t being Element of Bags n such that
A5: ( u9 = s + t & s in Support m ) and
A6: t in Support p ;
 u9 = (term m) + t by A1, A5, TARSKI:def_1;
hence  u in  {  ((term m) + b) where b is Element of Bags n : b in Support p  }  by A6; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in__{__((term_m)_+_b)_where_b_is_Element_of_Bags_n_:_b_in_Support_p__}__holds_
u_in_Support_(m_*'_p)
let u be set ; ::_thesis: ( u in  {  ((term m) + b) where b is Element of Bags n : b in Support p  }  implies u in Support (m *' p) )
assume  u in  {  ((term m) + b) where b is Element of Bags n : b in Support p  }  ; ::_thesis: u in Support (m *' p)
then  ex t being Element of Bags n st
( u = (term m) + t & t in Support p ) ;
hence  u in Support (m *' p) by Th8; ::_thesis: verum
end;
hence  Support (m *' p) =  {  ((term m) + b) where b is Element of Bags n : b in Support p  }  by A3, TARSKI:1; ::_thesis: verum
end;

theorem Th10: :: GROEB_3:10
for n being Ordinal
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds card (Support p) = card (Support (m *' p))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds card (Support p) = card (Support (m *' p))
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being non-zero Monomial of n,L holds card (Support p) = card (Support (m *' p))
let p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L holds card (Support p) = card (Support (m *' p))
let m be non-zero Monomial of n,L; ::_thesis: card (Support p) = card (Support (m *' p))
defpred S1[ set , set ] means $2 = (term m) + ((In ($1,(Bags n))) @);
set T = the connected admissible TermOrder of n;
 m <> 0_ (n,L) by POLYNOM7:def_1;
then  Support m <> {} by POLYNOM7:1;
then A1: Support m = {(term m)} by POLYNOM7:7;
A2: for x being set st x in Support p holds
ex y being set st
( y in Support (m *' p) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Support p implies ex y being set st
( y in Support (m *' p) & S1[x,y] ) )
assume A3: x in Support p ; ::_thesis: ex y being set st
( y in Support (m *' p) & S1[x,y] )
then reconsider x9 = x as Element of Bags n ;
A4: x9 = In (x9,(Bags n)) by FUNCT_7:def_1
.= (In (x9,(Bags n))) @ by POLYNOM2:def_3 ;
 (term m) + x9 in Support (m *' p) by A3, Th8;
hence  ex y being set st
( y in Support (m *' p) & S1[x,y] ) by A4; ::_thesis: verum
end;
consider f being Function of (Support p),(Support (m *' p)) such that
A5: for x being set st x in Support p holds
S1[x,f . x] from FUNCT_2:sch_1(A2);
A6: now__::_thesis:_(_Support_(m_*'_p)_=_{}_implies_Support_p_=_{}_)
assume A7: Support (m *' p) = {} ; ::_thesis: Support p = {}
now__::_thesis:_not_Support_p_<>_{}
assume  Support p <> {} ; ::_thesis: contradiction
then  p <> 0_ (n,L) by POLYNOM7:1;
then reconsider p9 = p as non-zero Polynomial of n,L by POLYNOM7:def_1;
 (HT (m, the connected admissible TermOrder of n)) + (HT (p9, the connected admissible TermOrder of n)) in Support (m *' p9) by TERMORD:29;
hence  contradiction by A7; ::_thesis: verum
end;
hence  Support p = {} ; ::_thesis: verum
end;
then A8: Support p c= dom f by FUNCT_2:def_1;
A9: Support (m *' p) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by TERMORD:30;
A10: now__::_thesis:_for_u_being_set_st_u_in_Support_(m_*'_p)_holds_
u_in_f_.:_(Support_p)
let u be set ; ::_thesis: ( u in Support (m *' p) implies u in f .: (Support p) )
assume A11: u in Support (m *' p) ; ::_thesis: u in f .: (Support p)
then reconsider u9 = u as Element of Bags n ;
 u9 in  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by A9, A11;
then consider s, t being Element of Bags n such that
A12: ( u9 = s + t & s in Support m ) and
A13: t in Support p ;
A14: t in dom f by A6, A13, FUNCT_2:def_1;
A15: t = In (t,(Bags n)) by FUNCT_7:def_1
.= (In (t,(Bags n))) @ by POLYNOM2:def_3 ;
 u9 = (term m) + t by A1, A12, TARSKI:def_1;
then  u9 = f . t by A5, A13, A15;
hence  u in f .: (Support p) by A14, FUNCT_1:def_6; ::_thesis: verum
end;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_Support_p_&_x2_in_Support_p_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in Support p & x2 in Support p & f . x1 = f . x2 implies x1 = x2 )
assume that
A16: x1 in Support p and
A17: x2 in Support p and
A18: f . x1 = f . x2 ; ::_thesis: x1 = x2
reconsider x19 = x1, x29 = x2 as Element of Bags n by A16, A17;
A19: x29 = In (x29,(Bags n)) by FUNCT_7:def_1
.= (In (x29,(Bags n))) @ by POLYNOM2:def_3 ;
x19 = In (x19,(Bags n)) by FUNCT_7:def_1
.= (In (x19,(Bags n))) @ by POLYNOM2:def_3 ;
then (term m) + x19 = f . x29 by A5, A16, A18
.= (term m) + x29 by A5, A17, A19 ;
hence x1 = (x29 + (term m)) -' (term m) by PRE_POLY:48
.= x2 by PRE_POLY:48 ;
::_thesis: verum
end;
then  f is one-to-one by A6, FUNCT_2:19;
then A20: Support p,f .: (Support p) are_equipotent by A8, CARD_1:33;
 for u being set st u in f .: (Support p) holds
u in Support (m *' p) ;
then  f .: (Support p) = Support (m *' p) by A10, TARSKI:1;
hence  card (Support p) = card (Support (m *' p)) by A20, CARD_1:5; ::_thesis: verum
end;

Lm1:  for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let f be Polynomial of n,L; ::_thesis: for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let g be set ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f,g implies g is Polynomial of n,L )
set R = PolyRedRel (P,T);
assume  PolyRedRel (P,T) reduces f,g ; ::_thesis: g is Polynomial of n,L
then consider p being RedSequence of PolyRedRel (P,T) such that
A1: p . 1 = f and
A2: p . (len p) = g by REWRITE1:def_3;
reconsider l = (len p) - 1 as Element of NAT by INT_1:5, NAT_1:14;
A3: 1 <= len p by NAT_1:14;
set h = p . l;
 1 <= l + 1 by NAT_1:12;
then A4: l + 1 in dom p by FINSEQ_3:25;
percases ( len p = 1 or len p <> 1 ) ;
suppose len p = 1 ; ::_thesis: g is Polynomial of n,L
hence  g is Polynomial of n,L by A1, A2; ::_thesis: verum
end;
suppose len p <> 1 ; ::_thesis: g is Polynomial of n,L
then  0 + 1 < l + 1 by A3, XXREAL_0:1;
then A5: 1 <= l by NAT_1:13;
 l <= l + 1 by NAT_1:13;
then  l in dom p by A5, FINSEQ_3:25;
then  [(p . l),g] in PolyRedRel (P,T) by A2, A4, REWRITE1:def_2;
then consider h9, g9 being set  such that
A6: [(p . l),g] = [h9,g9] and
 h9 in NonZero (Polynom-Ring (n,L)) and
A7: g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
 g = g9 by A6, XTUPLE_0:1;
hence  g is Polynomial of n,L by A7, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;

theorem Th11: :: GROEB_3:11
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr holds Red ((0_ (n,L)),T) = 0_ (n,L)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr holds Red ((0_ (n,L)),T) = 0_ (n,L)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr holds Red ((0_ (n,L)),T) = 0_ (n,L)
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: Red ((0_ (n,L)),T) = 0_ (n,L)
set e = 0_ (n,L);
set h = HM ((0_ (n,L)),T);
HC ((HM ((0_ (n,L)),T)),T) = HC ((0_ (n,L)),T) by TERMORD:27
.= (0_ (n,L)) . (HT ((0_ (n,L)),T)) by TERMORD:def_7
.= 0. L by POLYNOM1:22 ;
then  HM ((0_ (n,L)),T) = 0_ (n,L) by TERMORD:17;
hence  Red ((0_ (n,L)),T) = (0_ (n,L)) - (0_ (n,L)) by TERMORD:def_9
.= 0_ (n,L) by POLYRED:4 ;
::_thesis: verum
end;

theorem Th12: :: GROEB_3:12
for n being Ordinal
for L being non trivial right_complementable commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L st p - q = 0_ (n,L) holds
p = q
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L st p - q = 0_ (n,L) holds
p = q
let L be non trivial right_complementable commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L st p - q = 0_ (n,L) holds
p = q
let p, q be Polynomial of n,L; ::_thesis: ( p - q = 0_ (n,L) implies p = q )
assume  p - q = 0_ (n,L) ; ::_thesis: p = q
hence q = q + (p - q) by POLYNOM1:23
.= q + (p + (- q)) by POLYNOM1:def_6
.= (q + (- q)) + p by POLYNOM1:21
.= (0_ (n,L)) + p by POLYRED:3
.= p by POLYRED:2 ;
::_thesis: verum
end;

theorem Th13: :: GROEB_3:13
for X being set
for L being non empty right_complementable add-associative right_zeroed addLoopStr holds - (0_ (X,L)) = 0_ (X,L)
proof
let X be set ; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr holds - (0_ (X,L)) = 0_ (X,L)
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: - (0_ (X,L)) = 0_ (X,L)
set o = - (0_ (X,L));
set e = 0_ (X,L);
A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(-_(0__(X,L)))_holds_
(-_(0__(X,L)))_._x_=_(0__(X,L))_._x
let x be set ; ::_thesis: ( x in dom (- (0_ (X,L))) implies (- (0_ (X,L))) . x = (0_ (X,L)) . x )
assume  x in dom (- (0_ (X,L))) ; ::_thesis: (- (0_ (X,L))) . x = (0_ (X,L)) . x
then reconsider b = x as bag of X ;
(- (0_ (X,L))) . b = - ((0_ (X,L)) . b) by POLYNOM1:17
.= - (0. L) by POLYNOM1:22
.= 0. L by RLVECT_1:12
.= (0_ (X,L)) . b by POLYNOM1:22 ;
hence  (- (0_ (X,L))) . x = (0_ (X,L)) . x ; ::_thesis: verum
end;
dom (- (0_ (X,L))) = Bags X by FUNCT_2:def_1
.= dom (0_ (X,L)) by FUNCT_2:def_1 ;
hence  - (0_ (X,L)) = 0_ (X,L) by A1, FUNCT_1:2; ::_thesis: verum
end;

theorem Th14: :: GROEB_3:14
for X being set
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for f being Series of X,L holds (0_ (X,L)) - f = - f
proof
let X be set ; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for f being Series of X,L holds (0_ (X,L)) - f = - f
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for f being Series of X,L holds (0_ (X,L)) - f = - f
let f be Series of X,L; ::_thesis: (0_ (X,L)) - f = - f
set p = (0_ (X,L)) - f;
A1: now__::_thesis:_for_x_being_set_st_x_in_dom_((0__(X,L))_-_f)_holds_
((0__(X,L))_-_f)_._x_=_(-_f)_._x
let x be set ; ::_thesis: ( x in dom ((0_ (X,L)) - f) implies ((0_ (X,L)) - f) . x = (- f) . x )
assume  x in dom ((0_ (X,L)) - f) ; ::_thesis: ((0_ (X,L)) - f) . x = (- f) . x
then reconsider b = x as Element of Bags X ;
((0_ (X,L)) - f) . b = ((0_ (X,L)) + (- f)) . b by POLYNOM1:def_6
.= ((0_ (X,L)) . b) + ((- f) . b) by POLYNOM1:15
.= (0. L) + ((- f) . b) by POLYNOM1:22
.= (- f) . b by ALGSTR_1:def_2 ;
hence  ((0_ (X,L)) - f) . x = (- f) . x ; ::_thesis: verum
end;
dom ((0_ (X,L)) - f) = Bags X by FUNCT_2:def_1
.= dom (- f) by FUNCT_2:def_1 ;
hence  (0_ (X,L)) - f = - f by A1, FUNCT_1:2; ::_thesis: verum
end;

theorem Th15: :: GROEB_3:15
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds p - (Red (p,T)) = HM (p,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds p - (Red (p,T)) = HM (p,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds p - (Red (p,T)) = HM (p,T)
let L be non trivial right_complementable add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds p - (Red (p,T)) = HM (p,T)
let p be Polynomial of n,L; ::_thesis: p - (Red (p,T)) = HM (p,T)
thus p - (Red (p,T)) = ((HM (p,T)) + (Red (p,T))) - (Red (p,T)) by TERMORD:38
.= ((HM (p,T)) + (Red (p,T))) + (- (Red (p,T))) by POLYNOM1:def_6
.= (HM (p,T)) + ((Red (p,T)) + (- (Red (p,T)))) by POLYNOM1:21
.= (HM (p,T)) + (0_ (n,L)) by POLYRED:3
.= HM (p,T) by POLYNOM1:23 ; ::_thesis: verum
end;

registration
let n be Ordinal;
let L be non empty right_complementable add-associative right_zeroed addLoopStr ;
let p be Polynomial of n,L;
cluster Support p ->  finite ;
coherence
 Support p is finite by POLYNOM1:def_4;
end;

definition
let n be Ordinal;
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ;
let p, q be Polynomial of n,L;
:: original: {
redefine func{p,q} ->  Subset of (Polynom-Ring (n,L));
coherence
 {p,q} is Subset of (Polynom-Ring (n,L))
proof
now__::_thesis:_for_u_being_set_st_u_in_{p,q}_holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in {p,q} implies u in the carrier of (Polynom-Ring (n,L)) )
assume A1: u in {p,q} ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
now__::_thesis:_(_(_u_=_p_&_u_in_the_carrier_of_(Polynom-Ring_(n,L))_)_or_(_u_=_q_&_u_in_the_carrier_of_(Polynom-Ring_(n,L))_)_)
percases ( u = p or u = q ) by A1, TARSKI:def_2;
case u = p ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
hence  u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
case u = q ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
hence  u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence  u in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: verum
end;
hence  {p,q} is Subset of (Polynom-Ring (n,L)) by TARSKI:def_3; ::_thesis: verum
end;
end;

begin

definition
let X be set ;
let L be non empty ZeroStr ;
let s be Series of X,L;
let Y be Subset of (Bags X);
funcs | Y ->  Series of X,L equals :: GROEB_3:def 1
s +* (((Support s) \ Y) --> (0. L));
coherence
 s +* (((Support s) \ Y) --> (0. L)) is Series of X,L
proof
set m = ((Support s) \ Y) --> (0. L);
set r = s +* (((Support s) \ Y) --> (0. L));
A1: now__::_thesis:_for_x_being_set_st_x_in_Bags_X_holds_
(s_+*_(((Support_s)_\_Y)_-->_(0._L)))_._x_in_the_carrier_of_L
let x be set ; ::_thesis: ( x in Bags X implies (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L )
assume  x in Bags X ; ::_thesis: (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L
then reconsider x9 = x as Element of Bags X ;
now__::_thesis:_(_(_x9_in_dom_(((Support_s)_\_Y)_-->_(0._L))_&_(s_+*_(((Support_s)_\_Y)_-->_(0._L)))_._x_in_the_carrier_of_L_)_or_(_not_x_in_dom_(((Support_s)_\_Y)_-->_(0._L))_&_(s_+*_(((Support_s)_\_Y)_-->_(0._L)))_._x_in_the_carrier_of_L_)_)
percases ( x9 in dom (((Support s) \ Y) --> (0. L)) or not x in dom (((Support s) \ Y) --> (0. L)) ) ;
caseA2: x9 in dom (((Support s) \ Y) --> (0. L)) ; ::_thesis: (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L
then (s +* (((Support s) \ Y) --> (0. L))) . x9 = (((Support s) \ Y) --> (0. L)) . x9 by FUNCT_4:13
.= 0. L by A2, FUNCOP_1:7 ;
hence  (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L ; ::_thesis: verum
end;
case not x in dom (((Support s) \ Y) --> (0. L)) ; ::_thesis: (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L
then  (s +* (((Support s) \ Y) --> (0. L))) . x9 = s . x9 by FUNCT_4:11;
hence  (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L ; ::_thesis: verum
end;
end;
end;
hence  (s +* (((Support s) \ Y) --> (0. L))) . x in the carrier of L ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(Support_s)_\_Y_holds_
u_in_Bags_X
let u be set ; ::_thesis: ( u in (Support s) \ Y implies u in Bags X )
assume  u in (Support s) \ Y ; ::_thesis: u in Bags X
then  u in Support s by XBOOLE_0:def_5;
hence  u in Bags X ; ::_thesis: verum
end;
then A3: (Support s) \ Y c= Bags X by TARSKI:def_3;
 ( dom s = Bags X & dom (((Support s) \ Y) --> (0. L)) = (Support s) \ Y ) by FUNCT_2:def_1;
then  dom (s +* (((Support s) \ Y) --> (0. L))) = (Bags X) \/ ((Support s) \ Y) by FUNCT_4:def_1;
hence  s +* (((Support s) \ Y) --> (0. L)) is Series of X,L by A3, A1, FUNCT_2:3, XBOOLE_1:12; ::_thesis: verum
end;
end;

:: deftheorem  defines | GROEB_3:def_1_:_
 for X being set
for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds s | Y = s +* (((Support s) \ Y) --> (0. L));

Lm2:  for X being set
for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds Support (s | Y) c= Support s
proof
let X be set ; ::_thesis: for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds Support (s | Y) c= Support s
let L be non empty ZeroStr ; ::_thesis: for s being Series of X,L
for Y being Subset of (Bags X) holds Support (s | Y) c= Support s
let s be Series of X,L; ::_thesis: for Y being Subset of (Bags X) holds Support (s | Y) c= Support s
let Y be Subset of (Bags X); ::_thesis: Support (s | Y) c= Support s
set m = ((Support s) \ Y) --> (0. L);
set r = s +* (((Support s) \ Y) --> (0. L));
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Support (s | Y) or u in Support s )
assume A1: u in Support (s | Y) ; ::_thesis: u in Support s
then reconsider u9 = u as Element of Bags X ;
A2: (s +* (((Support s) \ Y) --> (0. L))) . u9 <> 0. L by A1, POLYNOM1:def_3;
now__::_thesis:_(_(_u9_in_dom_(((Support_s)_\_Y)_-->_(0._L))_&_u9_in_Support_s_)_or_(_not_u9_in_dom_(((Support_s)_\_Y)_-->_(0._L))_&_s_._u9_<>_0._L_)_)
percases ( u9 in dom (((Support s) \ Y) --> (0. L)) or not u9 in dom (((Support s) \ Y) --> (0. L)) ) ;
case u9 in dom (((Support s) \ Y) --> (0. L)) ; ::_thesis: u9 in Support s
hence  u9 in Support s by XBOOLE_0:def_5; ::_thesis: verum
end;
case not u9 in dom (((Support s) \ Y) --> (0. L)) ; ::_thesis: s . u9 <> 0. L
hence  s . u9 <> 0. L by A2, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
hence  u in Support s by POLYNOM1:def_3; ::_thesis: verum
end;

registration
let n be Ordinal;
let L be non empty ZeroStr ;
let p be Polynomial of n,L;
let Y be Subset of (Bags n);
clusterp | Y ->  finite-Support ;
coherence
 p | Y is finite-Support
proof
 Support p is finite by POLYNOM1:def_4;
then  Support (p | Y) is finite by Lm2, FINSET_1:1;
hence  p | Y is finite-Support by POLYNOM1:def_4; ::_thesis: verum
end;
end;

theorem Th16: :: GROEB_3:16
for X being set
for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds
( Support (s | Y) = (Support s) /\ Y & ( for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ) )
proof
let X be set ; ::_thesis: for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds
( Support (s | Y) = (Support s) /\ Y & ( for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ) )
let L be non empty ZeroStr ; ::_thesis: for s being Series of X,L
for Y being Subset of (Bags X) holds
( Support (s | Y) = (Support s) /\ Y & ( for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ) )
let s be Series of X,L; ::_thesis: for Y being Subset of (Bags X) holds
( Support (s | Y) = (Support s) /\ Y & ( for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ) )
let Y be Subset of (Bags X); ::_thesis: ( Support (s | Y) = (Support s) /\ Y & ( for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ) )
set m = ((Support s) \ Y) --> (0. L);
set r = s +* (((Support s) \ Y) --> (0. L));
A1: now__::_thesis:_for_u_being_set_st_u_in_Support_(s_|_Y)_holds_
u_in_(Support_s)_/\_Y
let u be set ; ::_thesis: ( u in Support (s | Y) implies u in (Support s) /\ Y )
assume A2: u in Support (s | Y) ; ::_thesis: u in (Support s) /\ Y
then reconsider u9 = u as Element of Bags X ;
A3: now__::_thesis:_not_u9_in_dom_(((Support_s)_\_Y)_-->_(0._L))
assume A4: u9 in dom (((Support s) \ Y) --> (0. L)) ; ::_thesis: contradiction
then (s +* (((Support s) \ Y) --> (0. L))) . u9 = (((Support s) \ Y) --> (0. L)) . u9 by FUNCT_4:13
.= 0. L by A4, FUNCOP_1:7 ;
hence  contradiction by A2, POLYNOM1:def_3; ::_thesis: verum
end;
 (s +* (((Support s) \ Y) --> (0. L))) . u9 <> 0. L by A2, POLYNOM1:def_3;
then  s . u9 <> 0. L by A3, FUNCT_4:11;
then A5: u9 in Support s by POLYNOM1:def_3;
 dom (((Support s) \ Y) --> (0. L)) = (Support s) \ Y by FUNCOP_1:13;
then  u9 in Y by A3, A5, XBOOLE_0:def_5;
hence  u in (Support s) /\ Y by A5, XBOOLE_0:def_4; ::_thesis: verum
end;
A6: dom (((Support s) \ Y) --> (0. L)) = (Support s) \ Y by FUNCOP_1:13;
now__::_thesis:_for_u_being_set_st_u_in_(Support_s)_/\_Y_holds_
u_in_Support_(s_|_Y)
let u be set ; ::_thesis: ( u in (Support s) /\ Y implies u in Support (s | Y) )
assume A7: u in (Support s) /\ Y ; ::_thesis: u in Support (s | Y)
then A8: u in Support s by XBOOLE_0:def_4;
then reconsider u9 = u as Element of Bags X ;
 u in Y by A7, XBOOLE_0:def_4;
then  not u in (Support s) \ Y by XBOOLE_0:def_5;
then  (s +* (((Support s) \ Y) --> (0. L))) . u9 = s . u9 by A6, FUNCT_4:11;
then  (s +* (((Support s) \ Y) --> (0. L))) . u9 <> 0. L by A8, POLYNOM1:def_3;
hence  u in Support (s | Y) by POLYNOM1:def_3; ::_thesis: verum
end;
hence A9: Support (s | Y) = (Support s) /\ Y by A1, TARSKI:1; ::_thesis: for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b
now__::_thesis:_for_b_being_bag_of_X_st_b_in_Support_(s_|_Y)_holds_
(s_|_Y)_._b_=_s_._b
let b be bag of X; ::_thesis: ( b in Support (s | Y) implies (s | Y) . b = s . b )
assume  b in Support (s | Y) ; ::_thesis: (s | Y) . b = s . b
then  b in Y by A9, XBOOLE_0:def_4;
then  not b in dom (((Support s) \ Y) --> (0. L)) by XBOOLE_0:def_5;
hence  (s | Y) . b = s . b by FUNCT_4:11; ::_thesis: verum
end;
hence  for b being bag of X st b in Support (s | Y) holds
(s | Y) . b = s . b ; ::_thesis: verum
end;

theorem :: GROEB_3:17
for X being set
for L being non empty ZeroStr
for s being Series of X,L
for Y being Subset of (Bags X) holds Support (s | Y) c= Support s by Lm2;

theorem :: GROEB_3:18
for X being set
for L being non empty ZeroStr
for s being Series of X,L holds
( s | (Support s) = s & s | ({} (Bags X)) = 0_ (X,L) )
proof
let X be set ; ::_thesis: for L being non empty ZeroStr
for s being Series of X,L holds
( s | (Support s) = s & s | ({} (Bags X)) = 0_ (X,L) )
let L be non empty ZeroStr ; ::_thesis: for s being Series of X,L holds
( s | (Support s) = s & s | ({} (Bags X)) = 0_ (X,L) )
let s be Series of X,L; ::_thesis: ( s | (Support s) = s & s | ({} (Bags X)) = 0_ (X,L) )
set r = s | (Support s);
set e = s | ({} (Bags X));
 ( s | (Support s) = s +* ({} --> (0. L)) & dom ({} --> (0. L)) = {} ) by XBOOLE_1:37;
hence s | (Support s) = s +* {}
.= s ;
::_thesis: s | ({} (Bags X)) = 0_ (X,L)
A1: dom ((Support s) --> (0. L)) = Support s by FUNCOP_1:13;
A2: now__::_thesis:_for_u_being_set_st_u_in_dom_(s_|_({}_(Bags_X)))_holds_
(s_|_({}_(Bags_X)))_._u_=_(0__(X,L))_._u
let u be set ; ::_thesis: ( u in dom (s | ({} (Bags X))) implies (s | ({} (Bags X))) . u = (0_ (X,L)) . u )
assume  u in dom (s | ({} (Bags X))) ; ::_thesis: (s | ({} (Bags X))) . u = (0_ (X,L)) . u
then reconsider u9 = u as Element of Bags X ;
now__::_thesis:_(_(_u9_in_Support_s_&_(s_|_({}_(Bags_X)))_._u9_=_(0__(X,L))_._u9_)_or_(_not_u9_in_Support_s_&_(s_|_({}_(Bags_X)))_._u9_=_(0__(X,L))_._u9_)_)
percases ( u9 in Support s or not u9 in Support s ) ;
caseA3: u9 in Support s ; ::_thesis: (s | ({} (Bags X))) . u9 = (0_ (X,L)) . u9
then (s | ({} (Bags X))) . u9 = ((Support s) --> (0. L)) . u9 by A1, FUNCT_4:13
.= 0. L by A3, FUNCOP_1:7 ;
hence  (s | ({} (Bags X))) . u9 = (0_ (X,L)) . u9 by POLYNOM1:22; ::_thesis: verum
end;
caseA4: not u9 in Support s ; ::_thesis: (s | ({} (Bags X))) . u9 = (0_ (X,L)) . u9
then  (s | ({} (Bags X))) . u9 = s . u9 by A1, FUNCT_4:11;
then  (s | ({} (Bags X))) . u9 = 0. L by A4, POLYNOM1:def_3;
hence  (s | ({} (Bags X))) . u9 = (0_ (X,L)) . u9 by POLYNOM1:22; ::_thesis: verum
end;
end;
end;
hence  (s | ({} (Bags X))) . u = (0_ (X,L)) . u ; ::_thesis: verum
end;
dom (s | ({} (Bags X))) = Bags X by FUNCT_2:def_1
.= dom (0_ (X,L)) by FUNCT_2:def_1 ;
hence  s | ({} (Bags X)) = 0_ (X,L) by A2, FUNCT_1:2; ::_thesis: verum
end;

definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty right_complementable add-associative right_zeroed addLoopStr ;
let p be Polynomial of n,L;
let i be Element of NAT ;
assume A1: i <= card (Support p) ;
func Upper_Support (p,T,i) ->  finite Subset of (Bags n) means :Def2: :: GROEB_3:def 2
( it c= Support p & card it = i & ( for b, b9 being bag of n st b in it & b9 in Support p & b <= b9,T holds
b9 in it ) );
existence
 ex b1 being finite Subset of (Bags n) st
( b1 c= Support p & card b1 = i & ( for b, b9 being bag of n st b in b1 & b9 in Support p & b <= b9,T holds
b9 in b1 ) )
proof
defpred S1[ Element of NAT ] means ( $1 > card (Support p) or ex M being finite Subset of (Bags n) st
( M c= Support p & card M = $1 & ( for b, b9 being bag of n st b in M & b9 in Support p & b <= b9,T holds
b9 in M ) ) );
A2: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
 ( k + 1 > card (Support p) or ex M being finite Subset of (Bags n) st
( M c= Support p & card M = k + 1 & ( for b, b9 being bag of n st b in M & b9 in Support p & b <= b9,T holds
b9 in M ) ) )
proof
set R = RelStr(# (Bags n),T #);
assume A4: not k + 1 > card (Support p) ; ::_thesis: ex M being finite Subset of (Bags n) st
( M c= Support p & card M = k + 1 & ( for b, b9 being bag of n st b in M & b9 in Support p & b <= b9,T holds
b9 in M ) )
 k <= k + 1 by NAT_1:11;
then consider M1 being finite Subset of (Bags n) such that
A5: M1 c= Support p and
A6: card M1 = k and
A7: for b, b9 being bag of n st b in M1 & b9 in Support p & b <= b9,T holds
b9 in M1 by A3, A4, XXREAL_0:2;
set G = (Support p) \ M1;
now__::_thesis:_for_u_being_set_st_u_in_(Support_p)_\_M1_holds_
u_in_Bags_n
let u be set ; ::_thesis: ( u in (Support p) \ M1 implies u in Bags n )
assume  u in (Support p) \ M1 ; ::_thesis: u in Bags n
then  u in Support p by XBOOLE_0:def_5;
hence  u in Bags n ; ::_thesis: verum
end;
then reconsider G = (Support p) \ M1 as Subset of (Bags n) by TARSKI:def_3;
A8: for u being set st u in M1 holds
u in Support p by A5;
now__::_thesis:_not_G_=_{}
assume  G = {} ; ::_thesis: contradiction
then  Support p c= M1 by XBOOLE_1:37;
then  for u being set st u in Support p holds
u in M1 ;
then  card (Support p) = k by A6, A8, TARSKI:1;
hence  contradiction by A4, NAT_1:16; ::_thesis: verum
end;
then reconsider G = G as non empty finite Subset of (Bags n) ;
consider x being Element of RelStr(# (Bags n),T #) such that
A9: x in G and
A10: x is_maximal_wrt G, the InternalRel of RelStr(# (Bags n),T #) by BAGORDER:6;
reconsider b = x as bag of n ;
set M = M1 \/ {b};
now__::_thesis:_for_u_being_set_st_u_in_{b}_holds_
u_in_Bags_n
let u be set ; ::_thesis: ( u in {b} implies u in Bags n )
assume  u in {b} ; ::_thesis: u in Bags n
then  u = b by TARSKI:def_1;
hence  u in Bags n ; ::_thesis: verum
end;
then  {b} c= Bags n by TARSKI:def_3;
then  M1 \/ {b} c= (Bags n) \/ (Bags n) by XBOOLE_1:13;
then reconsider M = M1 \/ {b} as finite Subset of (Bags n) ;
now__::_thesis:_for_u_being_set_st_u_in_M_holds_
u_in_Support_p
let u be set ; ::_thesis: ( u in M implies u in Support p )
assume A11: u in M ; ::_thesis: u in Support p
now__::_thesis:_(_(_u_in_M1_&_u_in_Support_p_)_or_(_u_in_{b}_&_u_in_Support_p_)_)
percases ( u in M1 or u in {b} ) by A11, XBOOLE_0:def_3;
case u in M1 ; ::_thesis: u in Support p
hence  u in Support p by A5; ::_thesis: verum
end;
case u in {b} ; ::_thesis: u in Support p
then  u in G by A9, TARSKI:def_1;
hence  u in Support p by XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
hence  u in Support p ; ::_thesis: verum
end;
then A12: M c= Support p by TARSKI:def_3;
A13: now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_G_holds_
b9_<=_b,T
let b9 be bag of n; ::_thesis: ( b9 in G implies b9 <= b,T )
assume A14: b9 in G ; ::_thesis: b9 <= b,T
now__::_thesis:_(_(_b9_=_b_&_b9_<=_b,T_)_or_(_b9_<>_b_&_b9_<=_b,T_)_)
percases ( b9 = b or b9 <> b ) ;
case b9 = b ; ::_thesis: b9 <= b,T
hence  b9 <= b,T by TERMORD:6; ::_thesis: verum
end;
case b9 <> b ; ::_thesis: b9 <= b,T
then  not [b,b9] in T by A10, A14, WAYBEL_4:def_23;
then  not b <= b9,T by TERMORD:def_2;
then  b9 < b,T by TERMORD:5;
hence  b9 <= b,T by TERMORD:def_3; ::_thesis: verum
end;
end;
end;
hence  b9 <= b,T ; ::_thesis: verum
end;
A15: now__::_thesis:_for_b1,_b2_being_bag_of_n_st_b1_in_M_&_b2_in_Support_p_&_b1_<=_b2,T_holds_
b2_in_M
let b1, b2 be bag of n; ::_thesis: ( b1 in M & b2 in Support p & b1 <= b2,T implies b2 in M )
assume that
A16: b1 in M and
A17: b2 in Support p and
A18: b1 <= b2,T ; ::_thesis: b2 in M
now__::_thesis:_(_(_b1_in_M1_&_b2_in_M_)_or_(_b1_in_{b}_&_b2_in_M_)_)
percases ( b1 in M1 or b1 in {b} ) by A16, XBOOLE_0:def_3;
case b1 in M1 ; ::_thesis: b2 in M
then  b2 in M1 by A7, A17, A18;
hence  b2 in M by XBOOLE_0:def_3; ::_thesis: verum
end;
case b1 in {b} ; ::_thesis: b2 in M
then A19: b1 = b by TARSKI:def_1;
now__::_thesis:_(_(_b2_=_b1_&_b2_in_M_)_or_(_b2_<>_b1_&_b2_in_M_)_)
percases ( b2 = b1 or b2 <> b1 ) ;
case b2 = b1 ; ::_thesis: b2 in M
hence  b2 in M by A16; ::_thesis: verum
end;
case b2 <> b1 ; ::_thesis: b2 in M
then A20: b1 < b2,T by A18, TERMORD:def_3;
now__::_thesis:_not_b2_in_G
assume  b2 in G ; ::_thesis: contradiction
then  b2 <= b1,T by A13, A19;
hence  contradiction by A20, TERMORD:5; ::_thesis: verum
end;
then  b2 in M1 by A17, XBOOLE_0:def_5;
hence  b2 in M by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence  b2 in M ; ::_thesis: verum
end;
end;
end;
hence  b2 in M ; ::_thesis: verum
end;
 not b in M1 by A9, XBOOLE_0:def_5;
then  card M = k + 1 by A6, CARD_2:41;
hence  ex M being finite Subset of (Bags n) st
( M c= Support p & card M = k + 1 & ( for b, b9 being bag of n st b in M & b9 in Support p & b <= b9,T holds
b9 in M ) ) by A12, A15; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
 ex M being finite Subset of (Bags n) st
( M c= Support p & card M = 0 & ( for b, b9 being bag of n st b in M & b9 in Support p & b <= b9,T holds
b9 in M ) )
proof
set M = {} (Bags n);
take  {} (Bags n) ; ::_thesis: ( {} (Bags n) c= Support p & card ({} (Bags n)) = 0 & ( for b, b9 being bag of n st b in {} (Bags n) & b9 in Support p & b <= b9,T holds
b9 in {} (Bags n) ) )
thus  ( {} (Bags n) c= Support p & card ({} (Bags n)) = 0 & ( for b, b9 being bag of n st b in {} (Bags n) & b9 in Support p & b <= b9,T holds
b9 in {} (Bags n) ) ) by XBOOLE_1:2; ::_thesis: verum
end;
then A21: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A21, A2);
hence  ex b1 being finite Subset of (Bags n) st
( b1 c= Support p & card b1 = i & ( for b, b9 being bag of n st b in b1 & b9 in Support p & b <= b9,T holds
b9 in b1 ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being finite Subset of (Bags n) st b1 c= Support p & card b1 = i & ( for b, b9 being bag of n st b in b1 & b9 in Support p & b <= b9,T holds
b9 in b1 ) & b2 c= Support p & card b2 = i & ( for b, b9 being bag of n st b in b2 & b9 in Support p & b <= b9,T holds
b9 in b2 ) holds
b1 = b2
proof
let F1, F2 be finite Subset of (Bags n); ::_thesis: ( F1 c= Support p & card F1 = i & ( for b, b9 being bag of n st b in F1 & b9 in Support p & b <= b9,T holds
b9 in F1 ) & F2 c= Support p & card F2 = i & ( for b, b9 being bag of n st b in F2 & b9 in Support p & b <= b9,T holds
b9 in F2 ) implies F1 = F2 )
assume that
A22: F1 c= Support p and
A23: card F1 = i and
A24: for b, b9 being bag of n st b in F1 & b9 in Support p & b <= b9,T holds
b9 in F1 ; ::_thesis: ( not F2 c= Support p or not card F2 = i or ex b, b9 being bag of n st
( b in F2 & b9 in Support p & b <= b9,T & not b9 in F2 ) or F1 = F2 )
assume that
A25: F2 c= Support p and
A26: card F2 = i and
A27: for b, b9 being bag of n st b in F2 & b9 in Support p & b <= b9,T holds
b9 in F2 ; ::_thesis: F1 = F2
now__::_thesis:_for_u_being_set_st_u_in_F1_holds_
u_in_F2
let u be set ; ::_thesis: ( u in F1 implies u in F2 )
assume A28: u in F1 ; ::_thesis: u in F2
then reconsider u9 = u as Element of Bags n ;
now__::_thesis:_u9_in_F2
assume A29: not u9 in F2 ; ::_thesis: contradiction
now__::_thesis:_for_x_being_set_st_x_in_F2_holds_
x_in_F1
let x be set ; ::_thesis: ( x in F2 implies x in F1 )
assume A30: x in F2 ; ::_thesis: x in F1
then reconsider x9 = x as Element of Bags n ;
now__::_thesis:_(_(_u9_<=_x9,T_&_x9_in_F1_)_or_(_not_u9_<=_x9,T_&_x9_in_F1_)_)
percases ( u9 <= x9,T or not u9 <= x9,T ) ;
case u9 <= x9,T ; ::_thesis: x9 in F1
hence  x9 in F1 by A24, A25, A28, A30; ::_thesis: verum
end;
case not u9 <= x9,T ; ::_thesis: x9 in F1
then  x9 < u9,T by TERMORD:5;
then  x9 <= u9,T by TERMORD:def_3;
hence  x9 in F1 by A22, A27, A28, A29, A30; ::_thesis: verum
end;
end;
end;
hence  x in F1 ; ::_thesis: verum
end;
then  F2 c= F1 by TARSKI:def_3;
then  F2 c< F1 by A28, A29, XBOOLE_0:def_8;
hence  contradiction by A23, A26, CARD_2:48; ::_thesis: verum
end;
hence  u in F2 ; ::_thesis: verum
end;
then  F1 c= F2 by TARSKI:def_3;
hence  F1 = F2 by A23, A26, PRE_POLY:8; ::_thesis: verum
end;
end;

:: deftheorem Def2 defines Upper_Support GROEB_3:def_2_:_
 for n being Ordinal
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 b6 being finite Subset of (Bags n) holds
( b6 = Upper_Support (p,T,i) iff ( b6 c= Support p & card b6 = i & ( for b, b9 being bag of n st b in b6 & b9 in Support p & b <= b9,T holds
b9 in b6 ) ) );

definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty right_complementable add-associative right_zeroed addLoopStr ;
let p be Polynomial of n,L;
let i be Element of NAT ;
func Lower_Support (p,T,i) ->  finite Subset of (Bags n) equals :: GROEB_3:def 3
(Support p) \ (Upper_Support (p,T,i));
coherence
 (Support p) \ (Upper_Support (p,T,i)) is finite Subset of (Bags n)
proof
 (Support p) \ (Upper_Support (p,T,i)) c= Support p by XBOOLE_1:36;
hence  (Support p) \ (Upper_Support (p,T,i)) is finite Subset of (Bags n) by XBOOLE_1:1; ::_thesis: verum
end;
end;

:: deftheorem  defines Lower_Support GROEB_3:def_3_:_
 for n being Ordinal
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 holds Lower_Support (p,T,i) = (Support p) \ (Upper_Support (p,T,i));

theorem Th19: :: GROEB_3:19
for n being Ordinal
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
( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} )
proof
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
( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_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 (Support p) holds
( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_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 (Support p) holds
( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_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 (Support p) holds
( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} )
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies ( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} ) )
set M = (Upper_Support (p,T,i)) /\ ((Support p) \ (Upper_Support (p,T,i)));
assume  i <= card (Support p) ; ::_thesis: ( (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = Support p & (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} )
then A1: Upper_Support (p,T,i) c= Support p by Def2;
thus (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) = (Upper_Support (p,T,i)) \/ (Support p) by XBOOLE_1:39
.= Support p by A1, XBOOLE_1:12 ; ::_thesis: (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {}
now__::_thesis:_not_(Upper_Support_(p,T,i))_/\_((Support_p)_\_(Upper_Support_(p,T,i)))_<>_{}
set x = the Element of (Upper_Support (p,T,i)) /\ ((Support p) \ (Upper_Support (p,T,i)));
assume  (Upper_Support (p,T,i)) /\ ((Support p) \ (Upper_Support (p,T,i))) <> {} ; ::_thesis: contradiction
then  ( the Element of (Upper_Support (p,T,i)) /\ ((Support p) \ (Upper_Support (p,T,i))) in Upper_Support (p,T,i) & the Element of (Upper_Support (p,T,i)) /\ ((Support p) \ (Upper_Support (p,T,i))) in (Support p) \ (Upper_Support (p,T,i)) ) by XBOOLE_0:def_4;
hence  contradiction by XBOOLE_0:def_5; ::_thesis: verum
end;
hence  (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} ; ::_thesis: verum
end;

theorem Th20: :: GROEB_3:20
for n being Ordinal
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, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
proof
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, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
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, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
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, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
for b, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies for b, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T )
assume A1: i <= card (Support p) ; ::_thesis: for b, b9 being bag of n st b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) holds
b9 < b,T
let b, b9 be bag of n; ::_thesis: ( b in Upper_Support (p,T,i) & b9 in Lower_Support (p,T,i) implies b9 < b,T )
assume that
A2: b in Upper_Support (p,T,i) and
A3: b9 in Lower_Support (p,T,i) ; ::_thesis: b9 < b,T
A4: Lower_Support (p,T,i) c= Support p by XBOOLE_1:36;
now__::_thesis:_not_b_<=_b9,T
assume  b <= b9,T ; ::_thesis: contradiction
then  b9 in Upper_Support (p,T,i) by A1, A2, A3, A4, Def2;
then  b9 in (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) by A3, XBOOLE_0:def_4;
hence  contradiction by A1, Th19; ::_thesis: verum
end;
hence  b9 < b,T by TERMORD:5; ::_thesis: verum
end;

theorem :: GROEB_3:21
for n being Ordinal
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 holds
( Upper_Support (p,T,0) = {} & Lower_Support (p,T,0) = Support p )
proof
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 holds
( Upper_Support (p,T,0) = {} & Lower_Support (p,T,0) = 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 holds
( Upper_Support (p,T,0) = {} & Lower_Support (p,T,0) = Support p )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L holds
( Upper_Support (p,T,0) = {} & Lower_Support (p,T,0) = Support p )
let p be Polynomial of n,L; ::_thesis: ( Upper_Support (p,T,0) = {} & Lower_Support (p,T,0) = Support p )
set u = Upper_Support (p,T,0);
 0 <= card (Support p) ;
then  card (Upper_Support (p,T,0)) = 0 by Def2;
hence  Upper_Support (p,T,0) = {} ; ::_thesis: Lower_Support (p,T,0) = Support p
hence  Lower_Support (p,T,0) = Support p ; ::_thesis: verum
end;

theorem Th22: :: GROEB_3:22
for n being Ordinal
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 holds
( Upper_Support (p,T,(card (Support p))) = Support p & Lower_Support (p,T,(card (Support p))) = {} )
proof
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 holds
( Upper_Support (p,T,(card (Support p))) = Support p & Lower_Support (p,T,(card (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 holds
( Upper_Support (p,T,(card (Support p))) = Support p & Lower_Support (p,T,(card (Support p))) = {} )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L holds
( Upper_Support (p,T,(card (Support p))) = Support p & Lower_Support (p,T,(card (Support p))) = {} )
let p be Polynomial of n,L; ::_thesis: ( Upper_Support (p,T,(card (Support p))) = Support p & Lower_Support (p,T,(card (Support p))) = {} )
set u = Upper_Support (p,T,(card (Support p)));
 ( Upper_Support (p,T,(card (Support p))) c= Support p & card (Upper_Support (p,T,(card (Support p)))) = card (Support p) ) by Def2;
hence  Upper_Support (p,T,(card (Support p))) = Support p by PRE_POLY:8; ::_thesis: Lower_Support (p,T,(card (Support p))) = {}
hence  Lower_Support (p,T,(card (Support p))) = {} by XBOOLE_1:37; ::_thesis: verum
end;

theorem Th23: :: GROEB_3:23
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Upper_Support (p,T,i)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Upper_Support (p,T,i)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Upper_Support (p,T,i)
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being non-zero Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Upper_Support (p,T,i)
let p be non-zero Polynomial of n,L; ::_thesis: for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Upper_Support (p,T,i)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= card (Support p) implies HT (p,T) in Upper_Support (p,T,i) )
assume that
A1: 1 <= i and
A2: i <= card (Support p) ; ::_thesis: HT (p,T) in Upper_Support (p,T,i)
 p <> 0_ (n,L) by POLYNOM7:def_1;
then  Support p <> {} by POLYNOM7:1;
then A3: HT (p,T) in Support p by TERMORD:def_6;
set u = Upper_Support (p,T,i);
set x = the Element of Upper_Support (p,T,i);
A4: Upper_Support (p,T,i) <> {} by A1, A2, Def2, CARD_1:27;
then A5: the Element of Upper_Support (p,T,i) in Upper_Support (p,T,i) ;
then reconsider x9 = the Element of Upper_Support (p,T,i) as Element of Bags n ;
 Upper_Support (p,T,i) c= Support p by A2, Def2;
then  x9 <= HT (p,T),T by A5, TERMORD:def_6;
hence  HT (p,T) in Upper_Support (p,T,i) by A2, A4, A3, Def2; ::_thesis: verum
end;

theorem Th24: :: GROEB_3:24
for n being Ordinal
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
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )
proof
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
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in 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 (Support p) holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in 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 (Support p) holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies ( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) ) )
assume A1: i <= card (Support p) ; ::_thesis: ( Lower_Support (p,T,i) c= Support p & card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )
set l = Lower_Support (p,T,i);
thus  Lower_Support (p,T,i) c= Support p by XBOOLE_1:36; ::_thesis: ( card (Lower_Support (p,T,i)) = (card (Support p)) - i & ( for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ) )
 Upper_Support (p,T,i) c= Support p by A1, Def2;
hence  card (Lower_Support (p,T,i)) = (card (Support p)) - (card (Upper_Support (p,T,i))) by CARD_2:44
.= (card (Support p)) - i by A1, Def2 ;
::_thesis: for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i)
now__::_thesis:_for_b,_b9_being_bag_of_n_st_b_in_Lower_Support_(p,T,i)_&_b9_in_Support_p_&_b9_<=_b,T_holds_
b9_in_Lower_Support_(p,T,i)
let b, b9 be bag of n; ::_thesis: ( b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T implies b9 in Lower_Support (p,T,i) )
assume that
A2: b in Lower_Support (p,T,i) and
A3: b9 in Support p and
A4: b9 <= b,T ; ::_thesis: b9 in Lower_Support (p,T,i)
A5: b9 in (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) by A1, A3, Th19;
now__::_thesis:_b9_in_Lower_Support_(p,T,i)
assume  not b9 in Lower_Support (p,T,i) ; ::_thesis: contradiction
then  b9 in Upper_Support (p,T,i) by A5, XBOOLE_0:def_3;
then  b < b9,T by A1, A2, Th20;
hence  contradiction by A4, TERMORD:5; ::_thesis: verum
end;
hence  b9 in Lower_Support (p,T,i) ; ::_thesis: verum
end;
hence  for b, b9 being bag of n st b in Lower_Support (p,T,i) & b9 in Support p & b9 <= b,T holds
b9 in Lower_Support (p,T,i) ; ::_thesis: verum
end;

definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty right_complementable add-associative right_zeroed addLoopStr ;
let p be Polynomial of n,L;
let i be Element of NAT ;
func Up (p,T,i) ->  Polynomial of n,L equals :: GROEB_3:def 4
p | (Upper_Support (p,T,i));
coherence
 p | (Upper_Support (p,T,i)) is Polynomial of n,L ;
func Low (p,T,i) ->  Polynomial of n,L equals :: GROEB_3:def 5
p | (Lower_Support (p,T,i));
coherence
 p | (Lower_Support (p,T,i)) is Polynomial of n,L ;
end;

:: deftheorem  defines Up GROEB_3:def_4_:_
 for n being Ordinal
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 holds Up (p,T,i) = p | (Upper_Support (p,T,i));

:: deftheorem  defines Low GROEB_3:def_5_:_
 for n being Ordinal
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 holds Low (p,T,i) = p | (Lower_Support (p,T,i));

Lm3:  for n being Ordinal
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 (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) & Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) )
proof
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 (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 (Support p) 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 (Support p) 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 (Support p) 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 (Support p) 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 (Support p) ; ::_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))) = (Support p) /\ (Upper_Support (p,T,i)) by Th16;
hence  Support (p | (Upper_Support (p,T,i))) = Upper_Support (p,T,i) by A1, XBOOLE_1:28; ::_thesis: Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i)
 Support (p | (Lower_Support (p,T,i))) = (Support p) /\ (Lower_Support (p,T,i)) by Th16;
hence  Support (p | (Lower_Support (p,T,i))) = Lower_Support (p,T,i) by XBOOLE_1:28, XBOOLE_1:36; ::_thesis: verum
end;

theorem :: GROEB_3:25
for n being Ordinal
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)) = Upper_Support (p,T,i) & Support (Low (p,T,i)) = Lower_Support (p,T,i) ) by Lm3;

theorem Th26: :: GROEB_3:26
for n being Ordinal
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 )
proof
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
end;

theorem Th27: :: GROEB_3:27
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low (p,T,i)) c= Support (Red (p,T))
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low (p,T,i)) c= Support (Red (p,T))
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low (p,T,i)) c= Support (Red (p,T))
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L
for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low (p,T,i)) c= Support (Red (p,T))
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st 1 <= i & i <= card (Support p) holds
Support (Low (p,T,i)) c= Support (Red (p,T))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= card (Support p) implies Support (Low (p,T,i)) c= Support (Red (p,T)) )
assume that
A1: 1 <= i and
A2: i <= card (Support p) ; ::_thesis: Support (Low (p,T,i)) c= Support (Red (p,T))
 Support p <> {} by A1, A2;
then  p <> 0_ (n,L) by POLYNOM7:1;
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def_1;
set sl = Lower_Support (p,T,i);
A3: now__::_thesis:_not_HT_(p,T)_in_Lower_Support_(p,T,i)
assume A4: HT (p,T) in Lower_Support (p,T,i) ; ::_thesis: contradiction
 HT (p,T) in Upper_Support (p,T,i) by A1, A2, Th23;
then  HT (p,T) in (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) by A4, XBOOLE_0:def_4;
hence  contradiction by A2, Th19; ::_thesis: verum
end;
now__::_thesis:_not_{(HT_(p,T))}_/\_(Lower_Support_(p,T,i))_<>_{}
set u = the Element of {(HT (p,T))} /\ (Lower_Support (p,T,i));
assume  {(HT (p,T))} /\ (Lower_Support (p,T,i)) <> {} ; ::_thesis: contradiction
then  ( the Element of {(HT (p,T))} /\ (Lower_Support (p,T,i)) in {(HT (p,T))} & the Element of {(HT (p,T))} /\ (Lower_Support (p,T,i)) in Lower_Support (p,T,i) ) by XBOOLE_0:def_4;
hence  contradiction by A3, TARSKI:def_1; ::_thesis: verum
end;
then  {(HT (p,T))} misses Lower_Support (p,T,i) by XBOOLE_0:def_7;
then A5: (Lower_Support (p,T,i)) \ {(HT (p,T))} = Lower_Support (p,T,i) by XBOOLE_1:83
.= Support (Low (p,T,i)) by A2, Lm3 ;
 (Support (Low (p,T,i))) \ {(HT (p,T))} c= (Support p) \ {(HT (p,T))} by A2, Th26, XBOOLE_1:33;
then  (Support (Low (p,T,i))) \ {(HT (p,T))} c= Support (Red (p,T)) by TERMORD:36;
hence  Support (Low (p,T,i)) c= Support (Red (p,T)) by A2, A5, Lm3; ::_thesis: verum
end;

theorem Th28: :: GROEB_3:28
for n being Ordinal
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 n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) )
proof
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 n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (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 (Support p) holds
for b being bag of n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (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 (Support p) holds
for b being bag of n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) )
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 n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) )
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies for b being bag of n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) ) )
assume A1: i <= card (Support p) ; ::_thesis: for b being bag of n st b in Support p holds
( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) )
let b be bag of n; ::_thesis: ( b in Support p implies ( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) ) )
assume A2: b in Support p ; ::_thesis: ( ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) & not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) )
Support p = (Upper_Support (p,T,i)) \/ (Lower_Support (p,T,i)) by A1, Th19
.= (Support (Up (p,T,i))) \/ (Lower_Support (p,T,i)) by A1, Lm3
.= (Support (Up (p,T,i))) \/ (Support (Low (p,T,i))) by A1, Lm3 ;
hence  ( b in Support (Up (p,T,i)) or b in Support (Low (p,T,i)) ) by A2, XBOOLE_0:def_3; ::_thesis: not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i)))
 ( Support (Up (p,T,i)) = Upper_Support (p,T,i) & Support (Low (p,T,i)) = Lower_Support (p,T,i) ) by A1, Lm3;
hence  not b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) by A1, Th19; ::_thesis: verum
end;

theorem Th29: :: GROEB_3:29
for n being Ordinal
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, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
proof
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, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
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, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
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, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
for b, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies for b, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T )
assume A1: i <= card (Support p) ; ::_thesis: for b, b9 being bag of n st b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) holds
b < b9,T
let b, b9 be bag of n; ::_thesis: ( b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) implies b < b9,T )
assume A2: ( b in Support (Low (p,T,i)) & b9 in Support (Up (p,T,i)) ) ; ::_thesis: b < b9,T
 ( Support (Up (p,T,i)) = Upper_Support (p,T,i) & Support (Low (p,T,i)) = Lower_Support (p,T,i) ) by A1, Lm3;
hence  b < b9,T by A1, A2, Th20; ::_thesis: verum
end;

theorem Th30: :: GROEB_3:30
for n being Ordinal
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 1 <= i & i <= card (Support p) holds
HT (p,T) in Support (Up (p,T,i))
proof
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 1 <= i & i <= card (Support p) holds
HT (p,T) in Support (Up (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 1 <= i & i <= card (Support p) holds
HT (p,T) in Support (Up (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 1 <= i & i <= card (Support p) holds
HT (p,T) in Support (Up (p,T,i))
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st 1 <= i & i <= card (Support p) holds
HT (p,T) in Support (Up (p,T,i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= card (Support p) implies HT (p,T) in Support (Up (p,T,i)) )
assume that
A1: 1 <= i and
A2: i <= card (Support p) ; ::_thesis: HT (p,T) in Support (Up (p,T,i))
 Support p <> {} by A1, A2;
then A3: HT (p,T) in Support p by TERMORD:def_6;
set u = Up (p,T,i);
set x = the Element of Support (Up (p,T,i));
A4: Support (Up (p,T,i)) = Upper_Support (p,T,i) by A2, Lm3;
then  card (Support (Up (p,T,i))) <> 0 by A1, A2, Def2;
then A5: Support (Up (p,T,i)) <> {} ;
then A6: the Element of Support (Up (p,T,i)) in Support (Up (p,T,i)) ;
then reconsider x = the Element of Support (Up (p,T,i)) as Element of Bags n ;
 Support (Up (p,T,i)) c= Support p by A2, A4, Def2;
then  x <= HT (p,T),T by A6, TERMORD:def_6;
hence  HT (p,T) in Support (Up (p,T,i)) by A2, A4, A5, A3, Def2; ::_thesis: verum
end;

theorem Th31: :: GROEB_3:31
for n being Ordinal
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 n st b in Support (Low (p,T,i)) holds
( (Low (p,T,i)) . b = p . b & (Up (p,T,i)) . b = 0. L )
proof
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 n 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 n 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 n 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 n 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 n st b in Support (Low (p,T,i)) holds
( (Low (p,T,i)) . b = p . b & (Up (p,T,i)) . b = 0. L ) )
set l = Lower_Support (p,T,i);
assume A1: i <= card (Support p) ; ::_thesis: for b being bag of n st b in Support (Low (p,T,i)) holds
( (Low (p,T,i)) . b = p . b & (Up (p,T,i)) . b = 0. L )
then A2: (Lower_Support (p,T,i)) /\ (Upper_Support (p,T,i)) = {} by Th19;
let b be bag of n; ::_thesis: ( b in Support (Low (p,T,i)) implies ( (Low (p,T,i)) . b = p . b & (Up (p,T,i)) . b = 0. L ) )
assume A3: b in Support (Low (p,T,i)) ; ::_thesis: ( (Low (p,T,i)) . b = p . b & (Up (p,T,i)) . b = 0. L )
hence  (Low (p,T,i)) . b = p . b by Th16; ::_thesis: (Up (p,T,i)) . b = 0. L
 b in Lower_Support (p,T,i) by A1, A3, Lm3;
then  not b in Upper_Support (p,T,i) by A2, XBOOLE_0:def_4;
then A4: not b in Support (Up (p,T,i)) by A1, Lm3;
 b is Element of Bags n by PRE_POLY:def_12;
hence  (Up (p,T,i)) . b = 0. L by A4, POLYNOM1:def_3; ::_thesis: verum
end;

theorem Th32: :: GROEB_3:32
for n being Ordinal
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 n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L )
proof
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 n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (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 n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (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 n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (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 n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L )
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies for b being bag of n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L ) )
set u = Upper_Support (p,T,i);
assume A1: i <= card (Support p) ; ::_thesis: for b being bag of n st b in Support (Up (p,T,i)) holds
( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L )
then A2: (Upper_Support (p,T,i)) /\ (Lower_Support (p,T,i)) = {} by Th19;
let b be bag of n; ::_thesis: ( b in Support (Up (p,T,i)) implies ( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L ) )
assume A3: b in Support (Up (p,T,i)) ; ::_thesis: ( (Up (p,T,i)) . b = p . b & (Low (p,T,i)) . b = 0. L )
hence  (Up (p,T,i)) . b = p . b by Th16; ::_thesis: (Low (p,T,i)) . b = 0. L
 b in Upper_Support (p,T,i) by A1, A3, Lm3;
then  not b in Lower_Support (p,T,i) by A2, XBOOLE_0:def_4;
then A4: not b in Support (Low (p,T,i)) by A1, Lm3;
 b is Element of Bags n by PRE_POLY:def_12;
hence  (Low (p,T,i)) . b = 0. L by A4, POLYNOM1:def_3; ::_thesis: verum
end;

theorem Th33: :: GROEB_3:33
for n being Ordinal
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
(Up (p,T,i)) + (Low (p,T,i)) = p
proof
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
(Up (p,T,i)) + (Low (p,T,i)) = 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
(Up (p,T,i)) + (Low (p,T,i)) = 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
(Up (p,T,i)) + (Low (p,T,i)) = p
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
(Up (p,T,i)) + (Low (p,T,i)) = p
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies (Up (p,T,i)) + (Low (p,T,i)) = p )
set u = (Up (p,T,i)) + (Low (p,T,i));
assume A1: i <= card (Support p) ; ::_thesis: (Up (p,T,i)) + (Low (p,T,i)) = p
A2: now__::_thesis:_for_x_being_set_st_x_in_Support_p_holds_
x_in_Support_((Up_(p,T,i))_+_(Low_(p,T,i)))
let x be set ; ::_thesis: ( x in Support p implies x in Support ((Up (p,T,i)) + (Low (p,T,i))) )
assume A3: x in Support p ; ::_thesis: x in Support ((Up (p,T,i)) + (Low (p,T,i)))
then reconsider x9 = x as Element of Bags n ;
A4: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = ((Up (p,T,i)) . x9) + ((Low (p,T,i)) . x9) by POLYNOM1:15;
A5: now__::_thesis:_(_(_x9_in_Support_(Up_(p,T,i))_&_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_=_p_._x9_)_or_(_x9_in_Support_(Low_(p,T,i))_&_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_=_p_._x9_)_)
percases ( x9 in Support (Up (p,T,i)) or x9 in Support (Low (p,T,i)) ) by A1, A3, Th28;
caseA6: x9 in Support (Up (p,T,i)) ; ::_thesis: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = p . x9
hence ((Up (p,T,i)) + (Low (p,T,i))) . x9 = ((Up (p,T,i)) . x9) + (0. L) by A1, A4, Th32
.= (Up (p,T,i)) . x9 by RLVECT_1:def_4
.= p . x9 by A1, A6, Th32 ;
::_thesis: verum
end;
caseA7: x9 in Support (Low (p,T,i)) ; ::_thesis: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = p . x9
hence ((Up (p,T,i)) + (Low (p,T,i))) . x9 = (0. L) + ((Low (p,T,i)) . x9) by A1, A4, Th31
.= (Low (p,T,i)) . x9 by ALGSTR_1:def_2
.= p . x9 by A1, A7, Th31 ;
::_thesis: verum
end;
end;
end;
 p . x9 <> 0. L by A3, POLYNOM1:def_3;
hence  x in Support ((Up (p,T,i)) + (Low (p,T,i))) by A5, POLYNOM1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_Support_((Up_(p,T,i))_+_(Low_(p,T,i)))_holds_
x_in_Support_p
let x be set ; ::_thesis: ( x in Support ((Up (p,T,i)) + (Low (p,T,i))) implies x in Support p )
 ( Support (Up (p,T,i)) c= Support p & Support (Low (p,T,i)) c= Support p ) by A1, Th26;
then  ( Support ((Up (p,T,i)) + (Low (p,T,i))) c= (Support (Up (p,T,i))) \/ (Support (Low (p,T,i))) & (Support (Up (p,T,i))) \/ (Support (Low (p,T,i))) c= Support p ) by POLYNOM1:20, XBOOLE_1:8;
then A8: Support ((Up (p,T,i)) + (Low (p,T,i))) c= Support p by XBOOLE_1:1;
assume  x in Support ((Up (p,T,i)) + (Low (p,T,i))) ; ::_thesis: x in Support p
hence  x in Support p by A8; ::_thesis: verum
end;
then A9: Support ((Up (p,T,i)) + (Low (p,T,i))) = Support p by A2, TARSKI:1;
A10: now__::_thesis:_for_x_being_set_st_x_in_dom_p_holds_
p_._x_=_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x
let x be set ; ::_thesis: ( x in dom p implies p . x = ((Up (p,T,i)) + (Low (p,T,i))) . x )
assume  x in dom p ; ::_thesis: p . x = ((Up (p,T,i)) + (Low (p,T,i))) . x
then reconsider x9 = x as Element of Bags n ;
A11: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = ((Up (p,T,i)) . x9) + ((Low (p,T,i)) . x9) by POLYNOM1:15;
now__::_thesis:_(_(_x9_in_Support_p_&_p_._x9_=_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_)_or_(_not_x9_in_Support_p_&_p_._x9_=_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_)_)
percases ( x9 in Support p or not x9 in Support p ) ;
caseA12: x9 in Support p ; ::_thesis: p . x9 = ((Up (p,T,i)) + (Low (p,T,i))) . x9
now__::_thesis:_(_(_x9_in_Support_(Up_(p,T,i))_&_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_=_p_._x9_)_or_(_x9_in_Support_(Low_(p,T,i))_&_((Up_(p,T,i))_+_(Low_(p,T,i)))_._x9_=_p_._x9_)_)
percases ( x9 in Support (Up (p,T,i)) or x9 in Support (Low (p,T,i)) ) by A1, A12, Th28;
caseA13: x9 in Support (Up (p,T,i)) ; ::_thesis: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = p . x9
hence ((Up (p,T,i)) + (Low (p,T,i))) . x9 = ((Up (p,T,i)) . x9) + (0. L) by A1, A11, Th32
.= (Up (p,T,i)) . x9 by RLVECT_1:def_4
.= p . x9 by A1, A13, Th32 ;
::_thesis: verum
end;
caseA14: x9 in Support (Low (p,T,i)) ; ::_thesis: ((Up (p,T,i)) + (Low (p,T,i))) . x9 = p . x9
hence ((Up (p,T,i)) + (Low (p,T,i))) . x9 = (0. L) + ((Low (p,T,i)) . x9) by A1, A11, Th31
.= (Low (p,T,i)) . x9 by ALGSTR_1:def_2
.= p . x9 by A1, A14, Th31 ;
::_thesis: verum
end;
end;
end;
hence  p . x9 = ((Up (p,T,i)) + (Low (p,T,i))) . x9 ; ::_thesis: verum
end;
caseA15: not x9 in Support p ; ::_thesis: p . x9 = ((Up (p,T,i)) + (Low (p,T,i))) . x9
hence p . x9 = 0. L by POLYNOM1:def_3
.= ((Up (p,T,i)) + (Low (p,T,i))) . x9 by A9, A15, POLYNOM1:def_3 ;
::_thesis: verum
end;
end;
end;
hence  p . x = ((Up (p,T,i)) + (Low (p,T,i))) . x ; ::_thesis: verum
end;
dom p = Bags n by FUNCT_2:def_1
.= dom ((Up (p,T,i)) + (Low (p,T,i))) by FUNCT_2:def_1 ;
hence  (Up (p,T,i)) + (Low (p,T,i)) = p by A10, FUNCT_1:2; ::_thesis: verum
end;

theorem Th34: :: GROEB_3:34
for n being Ordinal
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 holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
proof
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 holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = 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 holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L holds
( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
let p be Polynomial of n,L; ::_thesis: ( Up (p,T,0) = 0_ (n,L) & Low (p,T,0) = p )
set u = Up (p,T,0);
set l = Low (p,T,0);
A1: 0 <= card (Support p) ;
then  Support (Up (p,T,0)) = Upper_Support (p,T,0) by Lm3;
then  card (Support (Up (p,T,0))) = 0 by A1, Def2;
then  Support (Up (p,T,0)) = {} ;
hence  Up (p,T,0) = 0_ (n,L) by POLYNOM7:1; ::_thesis: Low (p,T,0) = p
then  (0_ (n,L)) + (Low (p,T,0)) = p by A1, Th33;
hence  Low (p,T,0) = p by POLYRED:2; ::_thesis: verum
end;

theorem Th35: :: GROEB_3:35
for n being Ordinal
for T being connected TermOrder of n
for L being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds
( Up (p,T,(card (Support p))) = p & Low (p,T,(card (Support p))) = 0_ (n,L) )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds
( Up (p,T,(card (Support p))) = p & Low (p,T,(card (Support p))) = 0_ (n,L) )
let T be connected TermOrder of n; ::_thesis: for L being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds
( Up (p,T,(card (Support p))) = p & Low (p,T,(card (Support p))) = 0_ (n,L) )
let L be non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds
( Up (p,T,(card (Support p))) = p & Low (p,T,(card (Support p))) = 0_ (n,L) )
let p be Polynomial of n,L; ::_thesis: ( Up (p,T,(card (Support p))) = p & Low (p,T,(card (Support p))) = 0_ (n,L) )
set u = Up (p,T,(card (Support p)));
set l = Low (p,T,(card (Support p)));
 Support (Up (p,T,(card (Support p)))) = (Support p) /\ (Upper_Support (p,T,(card (Support p)))) by Th16;
then A1: Support (Up (p,T,(card (Support p)))) c= Support p by XBOOLE_1:17;
A2: card (Support (Up (p,T,(card (Support p))))) = card (Upper_Support (p,T,(card (Support p)))) by Lm3
.= card (Support p) by Th22 ;
A3: now__::_thesis:_for_x_being_set_st_x_in_Support_p_holds_
x_in_Support_(Up_(p,T,(card_(Support_p))))
let x be set ; ::_thesis: ( x in Support p implies x in Support (Up (p,T,(card (Support p)))) )
assume A4: x in Support p ; ::_thesis: x in Support (Up (p,T,(card (Support p))))
now__::_thesis:_x_in_Support_(Up_(p,T,(card_(Support_p))))
assume  not x in Support (Up (p,T,(card (Support p)))) ; ::_thesis: contradiction
then  Support (Up (p,T,(card (Support p)))) c< Support p by A1, A4, XBOOLE_0:def_8;
hence  contradiction by A2, CARD_2:48; ::_thesis: verum
end;
hence  x in Support (Up (p,T,(card (Support p)))) ; ::_thesis: verum
end;
 for x being set st x in Support (Up (p,T,(card (Support p)))) holds
x in Support p by A1;
then A5: Support (Up (p,T,(card (Support p)))) = Support p by A3, TARSKI:1;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_p_holds_
p_._x_=_(Up_(p,T,(card_(Support_p))))_._x
let x be set ; ::_thesis: ( x in dom p implies p . x = (Up (p,T,(card (Support p)))) . x )
assume  x in dom p ; ::_thesis: p . x = (Up (p,T,(card (Support p)))) . x
then reconsider x9 = x as Element of Bags n ;
now__::_thesis:_(_(_x_in_Support_p_&_p_._x9_=_(Up_(p,T,(card_(Support_p))))_._x9_)_or_(_not_x_in_Support_p_&_p_._x9_=_(Up_(p,T,(card_(Support_p))))_._x9_)_)
percases ( x in Support p or not x in Support p ) ;
case x in Support p ; ::_thesis: p . x9 = (Up (p,T,(card (Support p)))) . x9
hence  p . x9 = (Up (p,T,(card (Support p)))) . x9 by A5, Th16; ::_thesis: verum
end;
caseA7: not x in Support p ; ::_thesis: p . x9 = (Up (p,T,(card (Support p)))) . x9
hence p . x9 = 0. L by POLYNOM1:def_3
.= (Up (p,T,(card (Support p)))) . x9 by A5, A7, POLYNOM1:def_3 ;
::_thesis: verum
end;
end;
end;
hence  p . x = (Up (p,T,(card (Support p)))) . x ; ::_thesis: verum
end;
dom p = Bags n by FUNCT_2:def_1
.= dom (Up (p,T,(card (Support p)))) by FUNCT_2:def_1 ;
hence A8: p = Up (p,T,(card (Support p))) by A6, FUNCT_1:2; ::_thesis: Low (p,T,(card (Support p))) = 0_ (n,L)
thus  0_ (n,L) = p + (- p) by POLYRED:3
.= ((Low (p,T,(card (Support p)))) + p) + (- p) by A8, Th33
.= (Low (p,T,(card (Support p)))) + (p + (- p)) by POLYNOM1:21
.= (Low (p,T,(card (Support p)))) + (0_ (n,L)) by POLYRED:3
.= Low (p,T,(card (Support p))) by POLYRED:2 ; ::_thesis: verum
end;

theorem Th36: :: GROEB_3:36
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being non-zero Polynomial of n,L holds
( Up (p,T,1) = HM (p,T) & Low (p,T,1) = Red (p,T) )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being non-zero Polynomial of n,L holds
( Up (p,T,1) = HM (p,T) & Low (p,T,1) = Red (p,T) )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable Abelian add-associative right_zeroed doubleLoopStr
for p being non-zero Polynomial of n,L holds
( Up (p,T,1) = HM (p,T) & Low (p,T,1) = Red (p,T) )
let L be non trivial right_complementable Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being non-zero Polynomial of n,L holds
( Up (p,T,1) = HM (p,T) & Low (p,T,1) = Red (p,T) )
let p be non-zero Polynomial of n,L; ::_thesis: ( Up (p,T,1) = HM (p,T) & Low (p,T,1) = Red (p,T) )
set u = Up (p,T,1);
set l = Low (p,T,1);
A1: now__::_thesis:_not_card_(Support_p)_<_1
assume  card (Support p) < 1 ; ::_thesis: contradiction
then  Support p = {} by NAT_1:14;
then  p = 0_ (n,L) by POLYNOM7:1;
hence  contradiction by POLYNOM7:def_1; ::_thesis: verum
end;
then  Support (Up (p,T,1)) = Upper_Support (p,T,1) by Lm3;
then  card (Support (Up (p,T,1))) = 1 by A1, Def2;
then consider x being set  such that
A2: Support (Up (p,T,1)) = {x} by CARD_2:42;
 HT (p,T) in {x} by A1, A2, Th30;
then A3: Support (Up (p,T,1)) = {(HT (p,T))} by A2, TARSKI:def_1;
 HM (p,T) <> 0_ (n,L) by POLYNOM7:def_1;
then  Support (HM (p,T)) <> {} by POLYNOM7:1;
then A4: Support (Up (p,T,1)) = Support (HM (p,T)) by A3, TERMORD:21;
A5: now__::_thesis:_for_x_being_set_st_x_in_dom_(HM_(p,T))_holds_
(HM_(p,T))_._x_=_(Up_(p,T,1))_._x
let x be set ; ::_thesis: ( x in dom (HM (p,T)) implies (HM (p,T)) . x = (Up (p,T,1)) . x )
assume  x in dom (HM (p,T)) ; ::_thesis: (HM (p,T)) . x = (Up (p,T,1)) . x
then reconsider x9 = x as Element of Bags n ;
now__::_thesis:_(_(_x_in_Support_(HM_(p,T))_&_(HM_(p,T))_._x9_=_(Up_(p,T,1))_._x9_)_or_(_not_x_in_Support_(HM_(p,T))_&_(HM_(p,T))_._x9_=_(Up_(p,T,1))_._x9_)_)
percases ( x in Support (HM (p,T)) or not x in Support (HM (p,T)) ) ;
caseA6: x in Support (HM (p,T)) ; ::_thesis: (HM (p,T)) . x9 = (Up (p,T,1)) . x9
then  x9 = HT (p,T) by A3, A4, TARSKI:def_1;
hence (HM (p,T)) . x9 = p . x9 by TERMORD:18
.= (Up (p,T,1)) . x9 by A4, A6, Th16 ;
::_thesis: verum
end;
caseA7: not x in Support (HM (p,T)) ; ::_thesis: (HM (p,T)) . x9 = (Up (p,T,1)) . x9
hence (HM (p,T)) . x9 = 0. L by POLYNOM1:def_3
.= (Up (p,T,1)) . x9 by A4, A7, POLYNOM1:def_3 ;
::_thesis: verum
end;
end;
end;
hence  (HM (p,T)) . x = (Up (p,T,1)) . x ; ::_thesis: verum
end;
dom (HM (p,T)) = Bags n by FUNCT_2:def_1
.= dom (Up (p,T,1)) by FUNCT_2:def_1 ;
hence  HM (p,T) = Up (p,T,1) by A5, FUNCT_1:2; ::_thesis: Low (p,T,1) = Red (p,T)
then A8: (HM (p,T)) + (Low (p,T,1)) = p by A1, Th33;
thus  Red (p,T) = p - (HM (p,T)) by TERMORD:def_9
.= ((Low (p,T,1)) + (HM (p,T))) + (- (HM (p,T))) by A8, POLYNOM1:def_6
.= (Low (p,T,1)) + ((HM (p,T)) + (- (HM (p,T)))) by POLYNOM1:21
.= (Low (p,T,1)) + (0_ (n,L)) by POLYRED:3
.= Low (p,T,1) by POLYRED:2 ; ::_thesis: verum
end;

registration
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ;
let p be non-zero Polynomial of n,L;
cluster Up (p,T,0) ->  monomial-like ;
coherence
 Up (p,T,0) is monomial-like
proof
 Up (p,T,0) = 0_ (n,L) by Th34;
hence  Up (p,T,0) is monomial-like ; ::_thesis: verum
end;
end;

registration
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable Abelian add-associative right_zeroed doubleLoopStr ;
let p be non-zero Polynomial of n,L;
cluster Up (p,T,1) ->  non-zero monomial-like ;
coherence
 ( Up (p,T,1) is non-zero & Up (p,T,1) is monomial-like )
proof
 Up (p,T,1) = HM (p,T) by Th36;
hence  ( Up (p,T,1) is non-zero & Up (p,T,1) is monomial-like ) ; ::_thesis: verum
end;
cluster Low (p,T,(card (Support p))) ->  monomial-like ;
coherence
 Low (p,T,(card (Support p))) is monomial-like
proof
 Low (p,T,(card (Support p))) = 0_ (n,L) by Th35;
hence  Low (p,T,(card (Support p))) is monomial-like ; ::_thesis: verum
end;
end;

theorem Th37: :: GROEB_3:37
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for j being Element of NAT st j = (card (Support p)) - 1 holds
Low (p,T,j) is non-zero Monomial of n,L
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for j being Element of NAT st j = (card (Support p)) - 1 holds
Low (p,T,j) is non-zero Monomial of n,L
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for j being Element of NAT st j = (card (Support p)) - 1 holds
Low (p,T,j) is non-zero Monomial of n,L
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L
for j being Element of NAT st j = (card (Support p)) - 1 holds
Low (p,T,j) is non-zero Monomial of n,L
let p be Polynomial of n,L; ::_thesis: for j being Element of NAT st j = (card (Support p)) - 1 holds
Low (p,T,j) is non-zero Monomial of n,L
let j be Element of NAT ; ::_thesis: ( j = (card (Support p)) - 1 implies Low (p,T,j) is non-zero Monomial of n,L )
set l = Low (p,T,j);
assume A1: j = (card (Support p)) - 1 ; ::_thesis: Low (p,T,j) is non-zero Monomial of n,L
A2: now__::_thesis:_not_j_>_card_(Support_p)
assume  j > card (Support p) ; ::_thesis: contradiction
then  ((card (Support p)) - 1) + 1 > (card (Support p)) + 1 by A1, XREAL_1:8;
then  (card (Support p)) + (- (card (Support p))) > ((card (Support p)) + 1) + (- (card (Support p))) by XREAL_1:8;
hence  contradiction ; ::_thesis: verum
end;
then  Support (Low (p,T,j)) = Lower_Support (p,T,j) by Lm3;
then  card (Support (Low (p,T,j))) = (card (Support p)) - ((card (Support p)) - 1) by A1, A2, Th24;
then consider x being set  such that
A3: Support (Low (p,T,j)) = {x} by CARD_2:42;
 x in Support (Low (p,T,j)) by A3, TARSKI:def_1;
then A4: x is Element of Bags n ;
 Low (p,T,j) <> 0_ (n,L) by A3, POLYNOM7:1;
hence  Low (p,T,j) is non-zero Monomial of n,L by A3, A4, POLYNOM7:6, POLYNOM7:def_1; ::_thesis: verum
end;

theorem Th38: :: GROEB_3:38
for n being Ordinal
for T being connected admissible 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
HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible 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
HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
let T be connected admissible 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
HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
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
HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i < card (Support p) holds
HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
let i be Element of NAT ; ::_thesis: ( i < card (Support p) implies HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T )
set li = Low (p,T,i);
set li1 = Low (p,T,(i + 1));
assume A1: i < card (Support p) ; ::_thesis: HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
then  Support (Low (p,T,i)) = Lower_Support (p,T,i) by Lm3;
then A2: card (Support (Low (p,T,i))) = (card (Support p)) - i by A1, Th24;
A3: i + 1 <= card (Support p) by A1, NAT_1:13;
then A4: Support (Low (p,T,(i + 1))) = Lower_Support (p,T,(i + 1)) by Lm3;
then A5: card (Support (Low (p,T,(i + 1)))) = (card (Support p)) - (i + 1) by A3, Th24;
A6: Support (Low (p,T,i)) c= Support p by A1, Th26;
now__::_thesis:_(_(_i_=_(card_(Support_p))_-_1_&_HT_((Low_(p,T,(i_+_1))),T)_<=_HT_((Low_(p,T,i)),T),T_)_or_(_i_<>_(card_(Support_p))_-_1_&_HT_((Low_(p,T,(i_+_1))),T)_<=_HT_((Low_(p,T,i)),T),T_)_)
percases ( i = (card (Support p)) - 1 or i <> (card (Support p)) - 1 ) ;
case i = (card (Support p)) - 1 ; ::_thesis: HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
then  card (Support (Low (p,T,(i + 1)))) = (card (Support p)) - (card (Support p)) by A4, Th24
.= 0 ;
then  Support (Low (p,T,(i + 1))) = {} ;
then  HT ((Low (p,T,(i + 1))),T) = EmptyBag n by TERMORD:def_6;
hence  HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T by TERMORD:9; ::_thesis: verum
end;
case i <> (card (Support p)) - 1 ; ::_thesis: HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T
then  card (Lower_Support (p,T,(i + 1))) <> 0 by A4, A5;
then  Lower_Support (p,T,(i + 1)) <> {} ;
then A7: HT ((Low (p,T,(i + 1))),T) in Lower_Support (p,T,(i + 1)) by A4, TERMORD:def_6;
now__::_thesis:_not_HT_((Low_(p,T,i)),T)_<_HT_((Low_(p,T,(i_+_1))),T),T
assume  HT ((Low (p,T,i)),T) < HT ((Low (p,T,(i + 1))),T),T ; ::_thesis: contradiction
then A8: HT ((Low (p,T,i)),T) <= HT ((Low (p,T,(i + 1))),T),T by TERMORD:def_3;
now__::_thesis:_for_u9_being_set_st_u9_in_Support_(Low_(p,T,i))_holds_
u9_in_Support_(Low_(p,T,(i_+_1)))
let u9 be set ; ::_thesis: ( u9 in Support (Low (p,T,i)) implies u9 in Support (Low (p,T,(i + 1))) )
assume A9: u9 in Support (Low (p,T,i)) ; ::_thesis: u9 in Support (Low (p,T,(i + 1)))
then reconsider u = u9 as Element of Bags n ;
 u <= HT ((Low (p,T,i)),T),T by A9, TERMORD:def_6;
hence  u9 in Support (Low (p,T,(i + 1))) by A3, A6, A4, A7, A8, A9, Th24, TERMORD:8; ::_thesis: verum
end;
then  Support (Low (p,T,i)) c= Support (Low (p,T,(i + 1))) by TARSKI:def_3;
then  (card (Support p)) + (- i) <= (card (Support p)) + (- (i + 1)) by A2, A5, NAT_1:43;
then  - i <= - (i + 1) by XREAL_1:6;
then  i + 1 <= i by XREAL_1:24;
then  (i + 1) - i <= i - i by XREAL_1:9;
then  1 <= 0 ;
hence  contradiction ; ::_thesis: verum
end;
hence  HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T by TERMORD:5; ::_thesis: verum
end;
end;
end;
hence  HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T ; ::_thesis: verum
end;

theorem Th39: :: GROEB_3:39
for n being Ordinal
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 0 < i & i < card (Support p) holds
HT ((Low (p,T,i)),T) < HT (p,T),T
proof
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 0 < i & i < card (Support p) holds
HT ((Low (p,T,i)),T) < HT (p,T),T
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 0 < i & i < card (Support p) holds
HT ((Low (p,T,i)),T) < HT (p,T),T
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 0 < i & i < card (Support p) holds
HT ((Low (p,T,i)),T) < HT (p,T),T
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st 0 < i & i < card (Support p) holds
HT ((Low (p,T,i)),T) < HT (p,T),T
let i be Element of NAT ; ::_thesis: ( 0 < i & i < card (Support p) implies HT ((Low (p,T,i)),T) < HT (p,T),T )
assume that
A1: 0 < i and
A2: i < card (Support p) ; ::_thesis: HT ((Low (p,T,i)),T) < HT (p,T),T
set l = Low (p,T,i);
now__::_thesis:_(_(_Low_(p,T,i)_=_0__(n,L)_&_contradiction_)_or_(_Low_(p,T,i)_<>_0__(n,L)_&_HT_((Low_(p,T,i)),T)_<_HT_(p,T),T_)_)
percases ( Low (p,T,i) = 0_ (n,L) or Low (p,T,i) <> 0_ (n,L) ) ;
case Low (p,T,i) = 0_ (n,L) ; ::_thesis: contradiction
then A3: card (Support (Low (p,T,i))) = 0 by CARD_1:27, POLYNOM7:1;
 Support (Low (p,T,i)) = Lower_Support (p,T,i) by A2, Lm3;
then  0 + i = ((card (Support p)) - i) + i by A2, A3, Th24;
hence  contradiction by A2; ::_thesis: verum
end;
caseA4: Low (p,T,i) <> 0_ (n,L) ; ::_thesis: HT ((Low (p,T,i)),T) < HT (p,T),T
A5: Support (Low (p,T,i)) c= Support p by A2, Th26;
A6: Support (Low (p,T,i)) = Lower_Support (p,T,i) by A2, Lm3;
A7: now__::_thesis:_not_HT_(p,T)_in_Support_(Low_(p,T,i))
assume A8: HT (p,T) in Support (Low (p,T,i)) ; ::_thesis: contradiction
A9: now__::_thesis:_for_u_being_set_st_u_in_Support_p_holds_
u_in_Support_(Low_(p,T,i))
let u be set ; ::_thesis: ( u in Support p implies u in Support (Low (p,T,i)) )
assume A10: u in Support p ; ::_thesis: u in Support (Low (p,T,i))
then reconsider x = u as Element of Bags n ;
 x <= HT (p,T),T by A10, TERMORD:def_6;
hence  u in Support (Low (p,T,i)) by A2, A6, A8, A10, Th24; ::_thesis: verum
end;
 for u being set st u in Support (Low (p,T,i)) holds
u in Support p by A5;
then  card (Support p) = card (Support (Low (p,T,i))) by A9, TARSKI:1
.= (card (Support p)) - i by A2, A6, Th24 ;
hence  contradiction by A1; ::_thesis: verum
end;
 Support (Low (p,T,i)) <> {} by A4, POLYNOM7:1;
then A11: HT ((Low (p,T,i)),T) in Support (Low (p,T,i)) by TERMORD:def_6;
then  HT ((Low (p,T,i)),T) <= HT (p,T),T by A5, TERMORD:def_6;
hence  HT ((Low (p,T,i)),T) < HT (p,T),T by A7, A11, TERMORD:def_3; ::_thesis: verum
end;
end;
end;
hence  HT ((Low (p,T,i)),T) < HT (p,T),T ; ::_thesis: verum
end;

theorem Th40: :: GROEB_3:40
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
let p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
let m be non-zero Monomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) ) )
set l = Low (p,T,i);
assume A1: i <= card (Support p) ; ::_thesis: for b being bag of n holds
( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
then A2: Support (Low (p,T,i)) c= Support p by Th26;
A3: Support (Up (p,T,i)) = Upper_Support (p,T,i) by A1, Lm3;
then A4: card (Support (Up (p,T,i))) = i by A1, Def2;
A5: Support (Up (p,T,i)) c= Support p by A1, Th26;
A6: Support (Low (p,T,i)) = Lower_Support (p,T,i) by A1, Lm3;
let b be bag of n; ::_thesis: ( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) )
A7: i <= card (Support (m *' p)) by A1, Th10;
then A8: Support (Low ((m *' p),T,i)) = Lower_Support ((m *' p),T,i) by Lm3;
A9: Support (Low ((m *' p),T,i)) c= Support (m *' p) by A7, Th26;
A10: Support (Up ((m *' p),T,i)) c= Support (m *' p) by A7, Th26;
A11: Support (Up ((m *' p),T,i)) = Upper_Support ((m *' p),T,i) by A7, Lm3;
then A12: card (Support (Up ((m *' p),T,i))) = i by A7, Def2;
then A13: ( Support (Up (p,T,i)) = {} implies Support (Up ((m *' p),T,i)) = {} ) by A4;
A14: now__::_thesis:_(_(term_m)_+_b_in_Support_(Low_((m_*'_p),T,i))_implies_b_in_Support_(Low_(p,T,i))_)
assume A15: (term m) + b in Support (Low ((m *' p),T,i)) ; ::_thesis: b in Support (Low (p,T,i))
A16: now__::_thesis:_not_(term_m)_+_b_in_Support_(Up_((m_*'_p),T,i))
assume  (term m) + b in Support (Up ((m *' p),T,i)) ; ::_thesis: contradiction
then  (term m) + b in (Support (Up ((m *' p),T,i))) /\ (Support (Low ((m *' p),T,i))) by A15, XBOOLE_0:def_4;
hence  contradiction by A7, A9, A15, Th28; ::_thesis: verum
end;
A17: Support (m *' p) =  {  ((term m) + u) where u is Element of Bags n : u in Support p  }  by Th9;
A18: now__::_thesis:_not_b_in_Support_(Up_(p,T,i))
defpred S1[ set , set ] means $1 = (term m) + ((In ($2,(Bags n))) @);
assume A19: b in Support (Up (p,T,i)) ; ::_thesis: contradiction
A20: now__::_thesis:_for_b9_being_bag_of_n_st_(term_m)_+_b9_in_Support_(Up_((m_*'_p),T,i))_holds_
b9_in_Support_(Up_(p,T,i))
let b9 be bag of n; ::_thesis: ( (term m) + b9 in Support (Up ((m *' p),T,i)) implies b9 in Support (Up (p,T,i)) )
assume A21: (term m) + b9 in Support (Up ((m *' p),T,i)) ; ::_thesis: b9 in Support (Up (p,T,i))
then A22: (term m) + b < (term m) + b9,T by A7, A11, A8, A15, Th20;
now__::_thesis:_not_b9_<=_b,T
assume  b9 <= b,T ; ::_thesis: contradiction
then  (term m) + b9 <= (term m) + b,T by Th2;
hence  contradiction by A22, TERMORD:5; ::_thesis: verum
end;
then  b < b9,T by TERMORD:5;
then A23: b <= b9,T by TERMORD:def_3;
 b9 in Support p by A10, A21, Th8;
hence  b9 in Support (Up (p,T,i)) by A1, A3, A19, A23, Def2; ::_thesis: verum
end;
A24: for x being set st x in Support (Up ((m *' p),T,i)) holds
ex y being set st
( y in Support (Up (p,T,i)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Support (Up ((m *' p),T,i)) implies ex y being set st
( y in Support (Up (p,T,i)) & S1[x,y] ) )
assume A25: x in Support (Up ((m *' p),T,i)) ; ::_thesis: ex y being set st
( y in Support (Up (p,T,i)) & S1[x,y] )
then  x in Support (m *' p) by A10;
then consider x9 being Element of Bags n such that
A26: x = (term m) + x9 and
 x9 in Support p by A17;
take  x9 ; ::_thesis: ( x9 in Support (Up (p,T,i)) & S1[x,x9] )
x9 = In (x9,(Bags n)) by FUNCT_7:def_1
.= (In (x9,(Bags n))) @ by POLYNOM2:def_3 ;
hence  ( x9 in Support (Up (p,T,i)) & S1[x,x9] ) by A20, A25, A26; ::_thesis: verum
end;
consider f being Function of (Support (Up ((m *' p),T,i))),(Support (Up (p,T,i))) such that
A27: for x being set st x in Support (Up ((m *' p),T,i)) holds
S1[x,f . x] from FUNCT_2:sch_1(A24);
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A28: x1 in dom f and
A29: x2 in dom f and
A30: f . x1 = f . x2 ; ::_thesis: x1 = x2
 f . x2 in rng f by A29, FUNCT_1:3;
then A31: f . x2 in Support (Up (p,T,i)) ;
 f . x1 in rng f by A28, FUNCT_1:3;
then  f . x1 in Support (Up (p,T,i)) ;
then reconsider x19 = f . x1, x29 = f . x2 as Element of Bags n by A31;
A32: x19 = In (x19,(Bags n)) by FUNCT_7:def_1
.= (In (x19,(Bags n))) @ by POLYNOM2:def_3 ;
x29 = In (x29,(Bags n)) by FUNCT_7:def_1
.= (In (x29,(Bags n))) @ by POLYNOM2:def_3 ;
hence x1 = (term m) + x19 by A27, A28, A30
.= x2 by A27, A29, A30, A32 ;
::_thesis: verum
end;
then A33: f is one-to-one by FUNCT_1:def_4;
 Support (Up ((m *' p),T,i)) c= dom f by A13, FUNCT_2:def_1;
then  Support (Up ((m *' p),T,i)),f .: (Support (Up ((m *' p),T,i))) are_equipotent by A33, CARD_1:33;
then  card (f .: (Support (Up ((m *' p),T,i)))) = card (Support (Up ((m *' p),T,i))) by CARD_1:5
.= i by A7, A11, Def2 ;
then  b in f .: (Support (Up ((m *' p),T,i))) by A4, A19, PRE_POLY:8;
then consider bb being set  such that
A34: bb in dom f and
A35: ( bb in Support (Up ((m *' p),T,i)) & f . bb = b ) by FUNCT_1:def_6;
 f . bb in rng f by A34, FUNCT_1:3;
then  f . bb in Support (Up (p,T,i)) ;
then reconsider bb9 = f . bb as Element of Bags n ;
bb9 = In (bb9,(Bags n)) by FUNCT_7:def_1
.= (In (bb9,(Bags n))) @ by POLYNOM2:def_3 ;
hence  contradiction by A16, A27, A35; ::_thesis: verum
end;
 (term m) + b in Support (m *' p) by A9, A15;
then  (term m) + b in  {  ((term m) + u) where u is Element of Bags n : u in Support p  }  by Th9;
then consider u being Element of Bags n such that
A36: (term m) + b = (term m) + u and
A37: u in Support p ;
b = ((term m) + b) -' (term m) by PRE_POLY:48
.= u by A36, PRE_POLY:48 ;
hence  b in Support (Low (p,T,i)) by A1, A37, A18, Th28; ::_thesis: verum
end;
A38: ( Support (Up ((m *' p),T,i)) = {} implies Support (Up (p,T,i)) = {} ) by A12, A4;
now__::_thesis:_(_b_in_Support_(Low_(p,T,i))_implies_(term_m)_+_b_in_Support_(Low_((m_*'_p),T,i))_)
assume A39: b in Support (Low (p,T,i)) ; ::_thesis: (term m) + b in Support (Low ((m *' p),T,i))
A40: now__::_thesis:_not_b_in_Support_(Up_(p,T,i))
assume  b in Support (Up (p,T,i)) ; ::_thesis: contradiction
then  b in (Support (Up (p,T,i))) /\ (Support (Low (p,T,i))) by A39, XBOOLE_0:def_4;
hence  contradiction by A1, A2, A39, Th28; ::_thesis: verum
end;
A41: now__::_thesis:_not_(term_m)_+_b_in_Support_(Up_((m_*'_p),T,i))
defpred S1[ set , set ] means $2 = (term m) + ((In ($1,(Bags n))) @);
assume A42: (term m) + b in Support (Up ((m *' p),T,i)) ; ::_thesis: contradiction
A43: now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_Support_(Up_(p,T,i))_holds_
(term_m)_+_b9_in_Support_(Up_((m_*'_p),T,i))
let b9 be bag of n; ::_thesis: ( b9 in Support (Up (p,T,i)) implies (term m) + b9 in Support (Up ((m *' p),T,i)) )
assume A44: b9 in Support (Up (p,T,i)) ; ::_thesis: (term m) + b9 in Support (Up ((m *' p),T,i))
then  (term m) + b < (term m) + b9,T by A1, A3, A6, A39, Th4, Th20;
then A45: (term m) + b <= (term m) + b9,T by TERMORD:def_3;
 (term m) + b9 in Support (m *' p) by A5, A44, Th8;
hence  (term m) + b9 in Support (Up ((m *' p),T,i)) by A7, A11, A42, A45, Def2; ::_thesis: verum
end;
A46: for x being set st x in Support (Up (p,T,i)) holds
ex y being set st
( y in Support (Up ((m *' p),T,i)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Support (Up (p,T,i)) implies ex y being set st
( y in Support (Up ((m *' p),T,i)) & S1[x,y] ) )
assume A47: x in Support (Up (p,T,i)) ; ::_thesis: ex y being set st
( y in Support (Up ((m *' p),T,i)) & S1[x,y] )
then reconsider x9 = x as Element of Bags n ;
take  (term m) + x9 ; ::_thesis: ( (term m) + x9 in Support (Up ((m *' p),T,i)) & S1[x,(term m) + x9] )
x9 = In (x9,(Bags n)) by FUNCT_7:def_1
.= (In (x9,(Bags n))) @ by POLYNOM2:def_3 ;
hence  ( (term m) + x9 in Support (Up ((m *' p),T,i)) & S1[x,(term m) + x9] ) by A43, A47; ::_thesis: verum
end;
consider f being Function of (Support (Up (p,T,i))),(Support (Up ((m *' p),T,i))) such that
A48: for x being set st x in Support (Up (p,T,i)) holds
S1[x,f . x] from FUNCT_2:sch_1(A46);
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A49: x1 in dom f and
A50: x2 in dom f and
A51: f . x1 = f . x2 ; ::_thesis: x1 = x2
 ( x1 in Support (Up (p,T,i)) & x2 in Support (Up (p,T,i)) ) by A49, A50;
then reconsider x = x1, y = x2 as Element of Bags n ;
y = In (y,(Bags n)) by FUNCT_7:def_1
.= (In (y,(Bags n))) @ by POLYNOM2:def_3 ;
then A52: f . y = (term m) + y by A48, A50;
x = In (x,(Bags n)) by FUNCT_7:def_1
.= (In (x,(Bags n))) @ by POLYNOM2:def_3 ;
then A53: f . x = (term m) + x by A48, A49;
thus x1 = ((term m) + x) -' (term m) by PRE_POLY:48
.= x2 by A51, A53, A52, PRE_POLY:48 ; ::_thesis: verum
end;
then A54: f is one-to-one by FUNCT_1:def_4;
 Support (Up (p,T,i)) c= dom f by A38, FUNCT_2:def_1;
then  Support (Up (p,T,i)),f .: (Support (Up (p,T,i))) are_equipotent by A54, CARD_1:33;
then  card (f .: (Support (Up (p,T,i)))) = card (Support (Up (p,T,i))) by CARD_1:5
.= i by A1, A3, Def2 ;
then  (term m) + b in f .: (Support (Up (p,T,i))) by A12, A42, PRE_POLY:8;
then consider bb being set  such that
A55: bb in dom f and
A56: bb in Support (Up (p,T,i)) and
A57: f . bb = (term m) + b by FUNCT_1:def_6;
reconsider bb = bb as Element of Bags n by A56;
bb = In (bb,(Bags n)) by FUNCT_7:def_1
.= (In (bb,(Bags n))) @ by POLYNOM2:def_3 ;
then A58: (term m) + bb = (term m) + b by A48, A55, A57;
bb = ((term m) + bb) -' (term m) by PRE_POLY:48
.= b by A58, PRE_POLY:48 ;
hence  contradiction by A40, A55; ::_thesis: verum
end;
 (term m) + b in Support (m *' p) by A2, A39, Th8;
hence  (term m) + b in Support (Low ((m *' p),T,i)) by A7, A41, Th28; ::_thesis: verum
end;
hence  ( (term m) + b in Support (Low ((m *' p),T,i)) iff b in Support (Low (p,T,i)) ) by A14; ::_thesis: verum
end;

theorem Th41: :: GROEB_3:41
for n being Ordinal
for T being connected admissible 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 (Low (p,T,(i + 1))) c= Support (Low (p,T,i))
proof
let n be Ordinal; ::_thesis: for T being connected admissible 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 (Low (p,T,(i + 1))) c= Support (Low (p,T,i))
let T be connected admissible 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 (Low (p,T,(i + 1))) c= Support (Low (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 (Support p) holds
Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i))
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i < card (Support p) holds
Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i))
let i be Element of NAT ; ::_thesis: ( i < card (Support p) implies Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i)) )
set l = Low (p,T,i);
set l1 = Low (p,T,(i + 1));
assume A1: i < card (Support p) ; ::_thesis: Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i))
then A2: i + 1 <= card (Support p) by NAT_1:13;
then A3: (card (Support p)) - i >= 1 by XREAL_1:19;
A4: Support (Low (p,T,i)) = Lower_Support (p,T,i) by A1, Lm3;
then  card (Support (Low (p,T,i))) = (card (Support p)) - i by A1, Th24;
then A5: HT ((Low (p,T,i)),T) in Lower_Support (p,T,i) by A3, A4, CARD_1:27, TERMORD:def_6;
A6: HT ((Low (p,T,(i + 1))),T) <= HT ((Low (p,T,i)),T),T by A1, Th38;
A7: Support (Low (p,T,(i + 1))) c= Support p by A2, Th26;
now__::_thesis:_for_u9_being_set_st_u9_in_Support_(Low_(p,T,(i_+_1)))_holds_
u9_in_Support_(Low_(p,T,i))
let u9 be set ; ::_thesis: ( u9 in Support (Low (p,T,(i + 1))) implies u9 in Support (Low (p,T,i)) )
assume A8: u9 in Support (Low (p,T,(i + 1))) ; ::_thesis: u9 in Support (Low (p,T,i))
then reconsider u = u9 as Element of Bags n ;
 u <= HT ((Low (p,T,(i + 1))),T),T by A8, TERMORD:def_6;
hence  u9 in Support (Low (p,T,i)) by A1, A7, A4, A6, A5, A8, Th24, TERMORD:8; ::_thesis: verum
end;
hence  Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i)) by TARSKI:def_3; ::_thesis: verum
end;

theorem Th42: :: GROEB_3:42
for n being Ordinal
for T being connected admissible 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 (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
proof
let n be Ordinal; ::_thesis: for T being connected admissible 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 (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
let T be connected admissible 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 (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
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 (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i < card (Support p) holds
(Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
let i be Element of NAT ; ::_thesis: ( i < card (Support p) implies (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))} )
set l = Low (p,T,i);
set l1 = Low (p,T,(i + 1));
assume A1: i < card (Support p) ; ::_thesis: (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))}
then A2: Support (Low (p,T,i)) = Lower_Support (p,T,i) by Lm3;
then A3: card (Support (Low (p,T,i))) = (card (Support p)) - i by A1, Th24;
now__::_thesis:_not_Lower_Support_(p,T,i)_=_{}
assume  Lower_Support (p,T,i) = {} ; ::_thesis: contradiction
then  (card (Support p)) - i = 0 by A1, Th24, CARD_1:27;
hence  contradiction by A1; ::_thesis: verum
end;
then A4: HT ((Low (p,T,i)),T) in Support (Low (p,T,i)) by A2, TERMORD:def_6;
A5: Support (Low (p,T,i)) c= Support p by A1, Th26;
A6: i + 1 <= card (Support p) by A1, NAT_1:13;
then  Support (Low (p,T,(i + 1))) = Lower_Support (p,T,(i + 1)) by Lm3;
then A7: card (Support (Low (p,T,(i + 1)))) = (card (Support p)) - (i + 1) by A6, Th24;
then  card ((Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1))))) = ((card (Support p)) - i) - ((card (Support p)) - (i + 1)) by A1, A3, Th41, CARD_2:44
.= 1 ;
then consider x being set  such that
A8: (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {x} by CARD_2:42;
A9: Support (Low (p,T,(i + 1))) = Lower_Support (p,T,(i + 1)) by A6, Lm3;
now__::_thesis:_not_x_<>_HT_((Low_(p,T,i)),T)
assume A10: x <> HT ((Low (p,T,i)),T) ; ::_thesis: contradiction
A11: now__::_thesis:_HT_((Low_(p,T,i)),T)_in_Support_(Low_(p,T,(i_+_1)))
assume  not HT ((Low (p,T,i)),T) in Support (Low (p,T,(i + 1))) ; ::_thesis: contradiction
then  HT ((Low (p,T,i)),T) in (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) by A4, XBOOLE_0:def_5;
hence  contradiction by A8, A10, TARSKI:def_1; ::_thesis: verum
end;
A12: now__::_thesis:_for_u_being_set_st_u_in_Support_(Low_(p,T,i))_holds_
u_in_Support_(Low_(p,T,(i_+_1)))
let u be set ; ::_thesis: ( u in Support (Low (p,T,i)) implies u in Support (Low (p,T,(i + 1))) )
assume A13: u in Support (Low (p,T,i)) ; ::_thesis: u in Support (Low (p,T,(i + 1)))
then reconsider u9 = u as Element of Bags n ;
 u9 <= HT ((Low (p,T,i)),T),T by A13, TERMORD:def_6;
hence  u in Support (Low (p,T,(i + 1))) by A6, A5, A9, A11, A13, Th24; ::_thesis: verum
end;
 Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i)) by A1, Th41;
then  for u being set st u in Support (Low (p,T,(i + 1))) holds
u in Support (Low (p,T,i)) ;
then  (card (Support p)) + (- i) <= (card (Support p)) + (- (i + 1)) by A3, A7, A12, TARSKI:1;
then  - i <= - (i + 1) by XREAL_1:6;
then  i + 1 <= i by XREAL_1:24;
then  (i + 1) - i <= i - i by XREAL_1:9;
then  1 <= 0 ;
hence  contradiction ; ::_thesis: verum
end;
hence  (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))} by A8; ::_thesis: verum
end;

theorem Th43: :: GROEB_3:43
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial 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
Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial 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
Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial 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
Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
let L be non trivial 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
Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
let p be Polynomial of n,L; ::_thesis: for i being Element of NAT st i < card (Support p) holds
Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
let i be Element of NAT ; ::_thesis: ( i < card (Support p) implies Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T) )
set l = Low (p,T,i);
set l1 = Low (p,T,(i + 1));
set r = Red ((Low (p,T,i)),T);
assume A1: i < card (Support p) ; ::_thesis: Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T)
then A2: Support (Low (p,T,i)) c= Support p by Th26;
 Support (Low (p,T,i)) = Lower_Support (p,T,i) by A1, Lm3;
then A3: card (Support (Low (p,T,i))) = (card (Support p)) - i by A1, Th24;
A4: Support (Low (p,T,(i + 1))) c= Support (Low (p,T,i)) by A1, Th41;
A5: i + 1 <= card (Support p) by A1, NAT_1:13;
then  Support (Low (p,T,(i + 1))) = Lower_Support (p,T,(i + 1)) by Lm3;
then A6: card (Support (Low (p,T,(i + 1)))) = (card (Support p)) - (i + 1) by A5, Th24;
A7: Support (Low (p,T,(i + 1))) = Lower_Support (p,T,(i + 1)) by A5, Lm3;
now__::_thesis:_not_{(HT_((Low_(p,T,i)),T))}_/\_(Support_(Low_(p,T,(i_+_1))))_<>_{}
set u = the Element of {(HT ((Low (p,T,i)),T))} /\ (Support (Low (p,T,(i + 1))));
assume A8: {(HT ((Low (p,T,i)),T))} /\ (Support (Low (p,T,(i + 1)))) <> {} ; ::_thesis: contradiction
then  the Element of {(HT ((Low (p,T,i)),T))} /\ (Support (Low (p,T,(i + 1)))) in {(HT ((Low (p,T,i)),T))} by XBOOLE_0:def_4;
then A9: the Element of {(HT ((Low (p,T,i)),T))} /\ (Support (Low (p,T,(i + 1)))) = HT ((Low (p,T,i)),T) by TARSKI:def_1;
A10: the Element of {(HT ((Low (p,T,i)),T))} /\ (Support (Low (p,T,(i + 1)))) in Support (Low (p,T,(i + 1))) by A8, XBOOLE_0:def_4;
now__::_thesis:_for_u9_being_set_st_u9_in_Support_(Low_(p,T,i))_holds_
u9_in_Support_(Low_(p,T,(i_+_1)))
let u9 be set ; ::_thesis: ( u9 in Support (Low (p,T,i)) implies u9 in Support (Low (p,T,(i + 1))) )
assume A11: u9 in Support (Low (p,T,i)) ; ::_thesis: u9 in Support (Low (p,T,(i + 1)))
then reconsider u = u9 as Element of Bags n ;
 u <= HT ((Low (p,T,i)),T),T by A11, TERMORD:def_6;
hence  u9 in Support (Low (p,T,(i + 1))) by A5, A2, A7, A10, A9, A11, Th24; ::_thesis: verum
end;
then  Support (Low (p,T,i)) c= Support (Low (p,T,(i + 1))) by TARSKI:def_3;
then  (card (Support p)) + (- i) <= (card (Support p)) + (- (i + 1)) by A3, A6, NAT_1:43;
then  - i <= - (i + 1) by XREAL_1:6;
then  i + 1 <= i by XREAL_1:24;
then  (i + 1) - i <= i - i by XREAL_1:9;
then  1 <= 0 ;
hence  contradiction ; ::_thesis: verum
end;
then A12: Support (Low (p,T,(i + 1))) misses {(HT ((Low (p,T,i)),T))} by XBOOLE_0:def_7;
A13: (Support (Low (p,T,i))) \ (Support (Low (p,T,(i + 1)))) = {(HT ((Low (p,T,i)),T))} by A1, Th42;
then  Support (Low (p,T,i)) = (Support (Low (p,T,(i + 1)))) \/ {(HT ((Low (p,T,i)),T))} by A1, Th41, XBOOLE_1:45;
then A14: Support (Red ((Low (p,T,i)),T)) = ((Support (Low (p,T,(i + 1)))) \/ {(HT ((Low (p,T,i)),T))}) \ {(HT ((Low (p,T,i)),T))} by TERMORD:36
.= (Support (Low (p,T,(i + 1)))) \ {(HT ((Low (p,T,i)),T))} by XBOOLE_1:40
.= Support (Low (p,T,(i + 1))) by A12, XBOOLE_1:83 ;
A15: now__::_thesis:_for_x_being_set_st_x_in_dom_(Low_(p,T,(i_+_1)))_holds_
(Low_(p,T,(i_+_1)))_._x_=_(Red_((Low_(p,T,i)),T))_._x
let x be set ; ::_thesis: ( x in dom (Low (p,T,(i + 1))) implies (Low (p,T,(i + 1))) . x = (Red ((Low (p,T,i)),T)) . x )
assume  x in dom (Low (p,T,(i + 1))) ; ::_thesis: (Low (p,T,(i + 1))) . x = (Red ((Low (p,T,i)),T)) . x
then reconsider b = x as Element of Bags n ;
now__::_thesis:_(_(_b_in_Support_(Low_(p,T,(i_+_1)))_&_(Low_(p,T,(i_+_1)))_._b_=_(Red_((Low_(p,T,i)),T))_._b_)_or_(_not_b_in_Support_(Low_(p,T,(i_+_1)))_&_(Low_(p,T,(i_+_1)))_._b_=_(Red_((Low_(p,T,i)),T))_._b_)_)
percases ( b in Support (Low (p,T,(i + 1))) or not b in Support (Low (p,T,(i + 1))) ) ;
caseA16: b in Support (Low (p,T,(i + 1))) ; ::_thesis: (Low (p,T,(i + 1))) . b = (Red ((Low (p,T,i)),T)) . b
then  not b in {(HT ((Low (p,T,i)),T))} by A13, XBOOLE_0:def_5;
then A17: b <> HT ((Low (p,T,i)),T) by TARSKI:def_1;
thus (Low (p,T,(i + 1))) . b = p . b by A5, A16, Th31
.= (Low (p,T,i)) . b by A1, A4, A16, Th31
.= (Red ((Low (p,T,i)),T)) . b by A4, A16, A17, TERMORD:40 ; ::_thesis: verum
end;
caseA18: not b in Support (Low (p,T,(i + 1))) ; ::_thesis: (Low (p,T,(i + 1))) . b = (Red ((Low (p,T,i)),T)) . b
hence (Low (p,T,(i + 1))) . b = 0. L by POLYNOM1:def_3
.= (Red ((Low (p,T,i)),T)) . b by A14, A18, POLYNOM1:def_3 ;
::_thesis: verum
end;
end;
end;
hence  (Low (p,T,(i + 1))) . x = (Red ((Low (p,T,i)),T)) . x ; ::_thesis: verum
end;
dom (Low (p,T,(i + 1))) = Bags n by FUNCT_2:def_1
.= dom (Red ((Low (p,T,i)),T)) by FUNCT_2:def_1 ;
hence  Low (p,T,(i + 1)) = Red ((Low (p,T,i)),T) by A15, FUNCT_1:2; ::_thesis: verum
end;

theorem Th44: :: GROEB_3:44
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
let p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L
for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
let m be non-zero Monomial of n,L; ::_thesis: for i being Element of NAT st i <= card (Support p) holds
Low ((m *' p),T,i) = m *' (Low (p,T,i))
let i be Element of NAT ; ::_thesis: ( i <= card (Support p) implies Low ((m *' p),T,i) = m *' (Low (p,T,i)) )
set l = Low (p,T,i);
set lm = Low ((m *' p),T,i);
assume A1: i <= card (Support p) ; ::_thesis: Low ((m *' p),T,i) = m *' (Low (p,T,i))
then A2: i <= card (Support (m *' p)) by Th10;
A3: Support (m *' (Low (p,T,i))) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support (Low (p,T,i)) )  }  by TERMORD:30;
A4: now__::_thesis:_for_u_being_set_st_u_in_Support_(m_*'_(Low_(p,T,i)))_holds_
u_in_Support_(Low_((m_*'_p),T,i))
 m <> 0_ (n,L) by POLYNOM7:def_1;
then  Support m <> {} by POLYNOM7:1;
then A5: Support m = {(term m)} by POLYNOM7:7;
then  term m in Support m by TARSKI:def_1;
then A6: m . (term m) <> 0. L by POLYNOM1:def_3;
let u be set ; ::_thesis: ( u in Support (m *' (Low (p,T,i))) implies u in Support (Low ((m *' p),T,i)) )
assume A7: u in Support (m *' (Low (p,T,i))) ; ::_thesis: u in Support (Low ((m *' p),T,i))
then reconsider u9 = u as Element of Bags n ;
 u in  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support (Low (p,T,i)) )  }  by A3, A7;
then consider s, t being Element of Bags n such that
A8: u9 = s + t and
A9: s in Support m and
A10: t in Support (Low (p,T,i)) ;
A11: (Low (p,T,i)) . t <> 0. L by A10, POLYNOM1:def_3;
A12: term m = s by A9, A5, TARSKI:def_1;
then (m *' p) . u9 = (m . (term m)) * (p . t) by A8, POLYRED:7
.= (m . (term m)) * ((Low (p,T,i)) . t) by A1, A10, Th31 ;
then  (m *' p) . u9 <> 0. L by A11, A6, VECTSP_2:def_1;
then A13: u9 in Support (m *' p) by POLYNOM1:def_3;
now__::_thesis:_s_+_t_in_Support_(Low_((m_*'_p),T,i))
assume  not s + t in Support (Low ((m *' p),T,i)) ; ::_thesis: contradiction
then A14: s + t in Support (Up ((m *' p),T,i)) by A2, A8, A13, Th28;
now__::_thesis:_for_t9_being_bag_of_n_st_t9_in_Support_(Low_(p,T,i))_holds_
t9_<_t,T
let t9 be bag of n; ::_thesis: ( t9 in Support (Low (p,T,i)) implies t9 < t,T )
assume  t9 in Support (Low (p,T,i)) ; ::_thesis: t9 < t,T
then  s + t9 in Support (Low ((m *' p),T,i)) by A1, A12, Th40;
then A15: s + t9 < s + t,T by A2, A14, Th29;
now__::_thesis:_not_t_<=_t9,T
assume  t <= t9,T ; ::_thesis: contradiction
then  s + t <= s + t9,T by Th2;
hence  contradiction by A15, TERMORD:5; ::_thesis: verum
end;
hence  t9 < t,T by TERMORD:5; ::_thesis: verum
end;
then  t < t,T by A10;
hence  contradiction by TERMORD:def_3; ::_thesis: verum
end;
hence  u in Support (Low ((m *' p),T,i)) by A8; ::_thesis: verum
end;
A16: Support (m *' p) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by TERMORD:30;
now__::_thesis:_for_u_being_set_st_u_in_Support_(Low_((m_*'_p),T,i))_holds_
u_in_Support_(m_*'_(Low_(p,T,i)))
let u be set ; ::_thesis: ( u in Support (Low ((m *' p),T,i)) implies u in Support (m *' (Low (p,T,i))) )
assume A17: u in Support (Low ((m *' p),T,i)) ; ::_thesis: u in Support (m *' (Low (p,T,i)))
then reconsider u9 = u as Element of Bags n ;
 Support (Low ((m *' p),T,i)) c= Support (m *' p) by A2, Th26;
then  u9 in Support (m *' p) by A17;
then A18: u9 in  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support p )  }  by A16;
 m <> 0_ (n,L) by POLYNOM7:def_1;
then  Support m <> {} by POLYNOM7:1;
then A19: Support m = {(term m)} by POLYNOM7:7;
then  term m in Support m by TARSKI:def_1;
then A20: m . (term m) <> 0. L by POLYNOM1:def_3;
consider s, t being Element of Bags n such that
A21: u = s + t and
A22: s in Support m and
A23: t in Support p by A18;
A24: p . t <> 0. L by A23, POLYNOM1:def_3;
A25: term m = s by A22, A19, TARSKI:def_1;
then A26: t in Support (Low (p,T,i)) by A1, A17, A21, Th40;
(m *' (Low (p,T,i))) . ((term m) + t) = (m . (term m)) * ((Low (p,T,i)) . t) by POLYRED:7
.= (m . (term m)) * (p . t) by A1, A26, Th31 ;
then  (m *' (Low (p,T,i))) . u9 <> 0. L by A21, A20, A25, A24, VECTSP_2:def_1;
hence  u in Support (m *' (Low (p,T,i))) by POLYNOM1:def_3; ::_thesis: verum
end;
then A27: Support (m *' (Low (p,T,i))) = Support (Low ((m *' p),T,i)) by A4, TARSKI:1;
A28: now__::_thesis:_for_x_being_set_st_x_in_dom_(m_*'_(Low_(p,T,i)))_holds_
(m_*'_(Low_(p,T,i)))_._x_=_(Low_((m_*'_p),T,i))_._x
let x be set ; ::_thesis: ( x in dom (m *' (Low (p,T,i))) implies (m *' (Low (p,T,i))) . x = (Low ((m *' p),T,i)) . x )
assume  x in dom (m *' (Low (p,T,i))) ; ::_thesis: (m *' (Low (p,T,i))) . x = (Low ((m *' p),T,i)) . x
then reconsider b = x as Element of Bags n ;
now__::_thesis:_(_(_b_in_Support_(m_*'_(Low_(p,T,i)))_&_(m_*'_(Low_(p,T,i)))_._b_=_(Low_((m_*'_p),T,i))_._b_)_or_(_not_b_in_Support_(m_*'_(Low_(p,T,i)))_&_(m_*'_(Low_(p,T,i)))_._b_=_(Low_((m_*'_p),T,i))_._b_)_)
percases ( b in Support (m *' (Low (p,T,i))) or not b in Support (m *' (Low (p,T,i))) ) ;
caseA29: b in Support (m *' (Low (p,T,i))) ; ::_thesis: (m *' (Low (p,T,i))) . b = (Low ((m *' p),T,i)) . b
then A30: b in  {  (s + t) where s, t is Element of Bags n : ( s in Support m & t in Support (Low (p,T,i)) )  }  by A3;
A31: b in Support (Low ((m *' p),T,i)) by A4, A29;
consider s, t being Element of Bags n such that
A32: b = s + t and
A33: s in Support m and
A34: t in Support (Low (p,T,i)) by A30;
 Support m = {(term m)} by A33, POLYNOM7:7;
then A35: term m = s by A33, TARSKI:def_1;
hence (m *' (Low (p,T,i))) . b = (m . (term m)) * ((Low (p,T,i)) . t) by A32, POLYRED:7
.= (m . (term m)) * (p . t) by A1, A34, Th31
.= (m *' p) . b by A32, A35, POLYRED:7
.= (Low ((m *' p),T,i)) . b by A2, A31, Th31 ;
::_thesis: verum
end;
caseA36: not b in Support (m *' (Low (p,T,i))) ; ::_thesis: (m *' (Low (p,T,i))) . b = (Low ((m *' p),T,i)) . b
hence (m *' (Low (p,T,i))) . b = 0. L by POLYNOM1:def_3
.= (Low ((m *' p),T,i)) . b by A27, A36, POLYNOM1:def_3 ;
::_thesis: verum
end;
end;
end;
hence  (m *' (Low (p,T,i))) . x = (Low ((m *' p),T,i)) . x ; ::_thesis: verum
end;
dom (m *' (Low (p,T,i))) = Bags n by FUNCT_2:def_1
.= dom (Low ((m *' p),T,i)) by FUNCT_2:def_1 ;
hence  Low ((m *' p),T,i) = m *' (Low (p,T,i)) by A28, FUNCT_1:2; ::_thesis: verum
end;

begin

Lm4:  for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for R being RedSequence of PolyRedRel (P,T)
for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for R being RedSequence of PolyRedRel (P,T)
for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for R being RedSequence of PolyRedRel (P,T)
for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for R being RedSequence of PolyRedRel (P,T)
for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for R being RedSequence of PolyRedRel (P,T)
for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
let R be RedSequence of PolyRedRel (P,T); ::_thesis: for i being Element of NAT st 1 <= i & i <= len R & len R > 1 holds
R . i is Polynomial of n,L
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len R & len R > 1 implies R . i is Polynomial of n,L )
assume that
A1: 1 <= i and
A2: i <= len R and
A3: 1 < len R ; ::_thesis: R . i is Polynomial of n,L
A4: i in dom R by A1, A2, FINSEQ_3:25;
now__::_thesis:_(_(_i_<>_len_R_&_R_._i_is_Polynomial_of_n,L_)_or_(_i_=_len_R_&_R_._i_is_Polynomial_of_n,L_)_)
percases ( i <> len R or i = len R ) ;
case i <> len R ; ::_thesis: R . i is Polynomial of n,L
then  i < len R by A2, XXREAL_0:1;
then  ( 1 <= i + 1 & i + 1 <= len R ) by NAT_1:11, NAT_1:13;
then  i + 1 in dom R by FINSEQ_3:25;
then  [(R . i),(R . (i + 1))] in PolyRedRel (P,T) by A4, REWRITE1:def_2;
then  R . i in dom (PolyRedRel (P,T)) by XTUPLE_0:def_12;
then  R . i in the carrier of (Polynom-Ring (n,L)) by XBOOLE_0:def_5;
hence  R . i is Polynomial of n,L by POLYNOM1:def_10; ::_thesis: verum
end;
caseA5: i = len R ; ::_thesis: R . i is Polynomial of n,L
A6: i - 1 is Element of NAT by A1, INT_1:5;
 1 + (- 1) < i + (- 1) by A3, A5, XREAL_1:8;
then A7: 1 <= i - 1 by A6, NAT_1:14;
A8: i = (i - 1) + 1 ;
 i - 1 <= len R by A5, XREAL_1:146;
then  i - 1 in dom R by A6, A7, FINSEQ_3:25;
then  [(R . (i - 1)),(R . i)] in PolyRedRel (P,T) by A4, A8, REWRITE1:def_2;
then  R . i in rng (PolyRedRel (P,T)) by XTUPLE_0:def_13;
hence  R . i is Polynomial of n,L by POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence  R . i is Polynomial of n,L ; ::_thesis: verum
end;

theorem Th45: :: GROEB_3:45
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
- f reduces_to - g,p,T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
- f reduces_to - g,p,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
- f reduces_to - g,p,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
- f reduces_to - g,p,T
let f, g, p be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies - f reduces_to - g,p,T )
assume  f reduces_to g,p,T ; ::_thesis: - f reduces_to - g,p,T
then consider b being bag of n such that
A1: f reduces_to g,p,b,T by POLYRED:def_6;
 b in Support f by A1, POLYRED:def_5;
then A2: b in Support (- f) by GROEB_1:5;
consider s being bag of n such that
A3: s + (HT (p,T)) = b and
A4: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A1, POLYRED:def_5;
 g = f + (- (((f . b) / (HC (p,T))) * (s *' p))) by A4, POLYNOM1:def_6;
then A5: - g = (- f) + (- (- (((f . b) / (HC (p,T))) * (s *' p)))) by POLYRED:1
.= (- f) - (- (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:def_6
.= (- f) - ((- ((f . b) / (HC (p,T)))) * (s *' p)) by POLYRED:9
.= (- f) - ((- ((f . b) * ((HC (p,T)) "))) * (s *' p)) by VECTSP_1:def_11
.= (- f) - (((- (f . b)) * ((HC (p,T)) ")) * (s *' p)) by VECTSP_1:9
.= (- f) - (((- (f . b)) / (HC (p,T))) * (s *' p)) by VECTSP_1:def_11
.= (- f) - ((((- f) . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:17 ;
A6: now__::_thesis:_not_-_f_=_0__(n,L)
a1: - (- f) = f by POLYNOM1:19;
assume  - f = 0_ (n,L) ; ::_thesis: contradiction
then f = - (0_ (n,L)) by a1
.= 0_ (n,L) by Th13 ;
hence  contradiction by A1, POLYRED:def_5; ::_thesis: verum
end;
 p <> 0_ (n,L) by A1, POLYRED:def_5;
then  - f reduces_to - g,p,b,T by A3, A6, A5, A2, POLYRED:def_5;
hence  - f reduces_to - g,p,T by POLYRED:def_6; ::_thesis: verum
end;

Lm5:  for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let p1, p2 be Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint implies for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 )
assume A1: HT (p1,T), HT (p2,T) are_disjoint ; ::_thesis: for b1, b2 being bag of n st b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let b1, b2 be bag of n; ::_thesis: ( b1 in Support p1 & b2 in Support p2 & ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) implies not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 )
assume that
A2: b1 in Support p1 and
A3: b2 in Support p2 ; ::_thesis: ( ( b1 = HT (p1,T) & b2 = HT (p2,T) ) or not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 )
assume A4: ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) ; ::_thesis: not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
 b2 <= HT (p2,T),T by A3, TERMORD:def_6;
then A5: (HT (p1,T)) + b2 <= (HT (p1,T)) + (HT (p2,T)),T by Th2;
 b1 <= HT (p1,T),T by A2, TERMORD:def_6;
then A6: (HT (p2,T)) + b1 <= (HT (p1,T)) + (HT (p2,T)),T by Th2;
assume A7: (HT (p1,T)) + b2 = (HT (p2,T)) + b1 ; ::_thesis: contradiction
then A8: HT (p1,T) divides (HT (p2,T)) + b1 by PRE_POLY:50;
A9: HT (p2,T) divides (HT (p1,T)) + b2 by A7, PRE_POLY:50;
now__::_thesis:_(_(_not_b1_=_HT_(p1,T)_&_contradiction_)_or_(_not_b2_=_HT_(p2,T)_&_contradiction_)_)
percases ( not b1 = HT (p1,T) or not b2 = HT (p2,T) ) by A4;
caseA10: not b1 = HT (p1,T) ; ::_thesis: contradiction
 HT (p2,T) divides (HT (p2,T)) + b1 by PRE_POLY:50;
then  lcm ((HT (p1,T)),(HT (p2,T))) divides (HT (p2,T)) + b1 by A8, GROEB_2:4;
then  (HT (p1,T)) + (HT (p2,T)) divides (HT (p2,T)) + b1 by A1, GROEB_2:5;
then  (HT (p1,T)) + (HT (p2,T)) <= (HT (p2,T)) + b1,T by TERMORD:10;
then A11: (HT (p1,T)) + (HT (p2,T)) = (HT (p2,T)) + b1 by A6, TERMORD:7;
 HT (p1,T) = ((HT (p1,T)) + (HT (p2,T))) -' (HT (p2,T)) by PRE_POLY:48;
hence  contradiction by A10, A11, PRE_POLY:48; ::_thesis: verum
end;
caseA12: not b2 = HT (p2,T) ; ::_thesis: contradiction
 HT (p1,T) divides (HT (p1,T)) + b2 by PRE_POLY:50;
then  lcm ((HT (p1,T)),(HT (p2,T))) divides (HT (p1,T)) + b2 by A9, GROEB_2:4;
then  (HT (p1,T)) + (HT (p2,T)) divides (HT (p1,T)) + b2 by A1, GROEB_2:5;
then  (HT (p1,T)) + (HT (p2,T)) <= (HT (p1,T)) + b2,T by TERMORD:10;
then A13: (HT (p1,T)) + (HT (p2,T)) = (HT (p1,T)) + b2 by A5, TERMORD:7;
 HT (p2,T) = ((HT (p1,T)) + (HT (p2,T))) -' (HT (p1,T)) by PRE_POLY:48;
hence  contradiction by A12, A13, PRE_POLY:48; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th46: :: GROEB_3:46
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st f reduces_to f1,{p},T & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
f + g reduces_to f1 + g,{p},T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st f reduces_to f1,{p},T & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
f + g reduces_to f1 + g,{p},T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st f reduces_to f1,{p},T & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
f + g reduces_to f1 + g,{p},T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, f1, g, p being Polynomial of n,L st f reduces_to f1,{p},T & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
f + g reduces_to f1 + g,{p},T
let f, f1, g, p be Polynomial of n,L; ::_thesis: ( f reduces_to f1,{p},T & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) implies f + g reduces_to f1 + g,{p},T )
assume that
A1: f reduces_to f1,{p},T and
A2: for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ; ::_thesis: f + g reduces_to f1 + g,{p},T
consider q being Polynomial of n,L such that
A3: q in {p} and
A4: f reduces_to f1,q,T by A1, POLYRED:def_7;
 p = q by A3, TARSKI:def_1;
then consider br being bag of n such that
A5: f reduces_to f1,p,br,T by A4, POLYRED:def_6;
consider s being bag of n such that
A6: s + (HT (p,T)) = br and
A7: f1 = f - (((f . br) / (HC (p,T))) * (s *' p)) by A5, POLYRED:def_5;
A8: now__::_thesis:_not_br_in_Support_g
assume A9: br in Support g ; ::_thesis: contradiction
 HT (p,T) divides br by A6, TERMORD:1;
hence  contradiction by A2, A9; ::_thesis: verum
end;
A10: br is Element of Bags n by PRE_POLY:def_12;
A11: p in {p} by TARSKI:def_1;
A12: br in Support f by A5, POLYRED:def_5;
A13: (f + g) . br = (f . br) + (g . br) by POLYNOM1:15
.= (f . br) + (0. L) by A8, A10, POLYNOM1:def_3
.= f . br by RLVECT_1:def_4 ;
A14: p <> 0_ (n,L) by A5, POLYRED:def_5;
now__::_thesis:_(_(_f_+_g_=_0__(n,L)_&_contradiction_)_or_(_f_+_g_<>_0__(n,L)_&_f_+_g_reduces_to_f1_+_g,{p},T_)_)
percases ( f + g = 0_ (n,L) or f + g <> 0_ (n,L) ) ;
case f + g = 0_ (n,L) ; ::_thesis: contradiction
then  (f + g) - f = - f by Th14;
then  (f + g) + (- f) = - f by POLYNOM1:def_6;
then  (f + (- f)) + g = - f by POLYNOM1:21;
then  (0_ (n,L)) + g = - f by POLYRED:3;
then  g = - f by POLYRED:2;
hence  contradiction by A12, A8, GROEB_1:5; ::_thesis: verum
end;
caseA15: f + g <> 0_ (n,L) ; ::_thesis: f + g reduces_to f1 + g,{p},T
set g1 = (f + g) - ((((f + g) . br) / (HC (p,T))) * (s *' p));
 (f + g) . br <> 0. L by A12, A13, POLYNOM1:def_3;
then  br in Support (f + g) by A12, POLYNOM1:def_3;
then  f + g reduces_to (f + g) - ((((f + g) . br) / (HC (p,T))) * (s *' p)),p,br,T by A14, A6, A15, POLYRED:def_5;
then A16: f + g reduces_to (f + g) - ((((f + g) . br) / (HC (p,T))) * (s *' p)),p,T by POLYRED:def_6;
(f + g) - ((((f + g) . br) / (HC (p,T))) * (s *' p)) = (f + g) + (- (((f . br) / (HC (p,T))) * (s *' p))) by A13, POLYNOM1:def_6
.= (f + (- (((f . br) / (HC (p,T))) * (s *' p)))) + g by POLYNOM1:21
.= f1 + g by A7, POLYNOM1:def_6 ;
hence  f + g reduces_to f1 + g,{p},T by A11, A16, POLYRED:def_7; ::_thesis: verum
end;
end;
end;
hence  f + g reduces_to f1 + g,{p},T ; ::_thesis: verum
end;

theorem Th47: :: GROEB_3:47
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds f *' g reduces_to (Red (f,T)) *' g,{g},T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds f *' g reduces_to (Red (f,T)) *' g,{g},T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds f *' g reduces_to (Red (f,T)) *' g,{g},T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being non-zero Polynomial of n,L holds f *' g reduces_to (Red (f,T)) *' g,{g},T
let f, g be non-zero Polynomial of n,L; ::_thesis: f *' g reduces_to (Red (f,T)) *' g,{g},T
set fg = f *' g;
set q = (f *' g) - ((((f *' g) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g));
reconsider r = - (HM (f,T)) as Polynomial of n,L ;
A1: g <> 0_ (n,L) by POLYNOM7:def_1;
A2: HC (g,T) <> 0. L ;
A3: f *' g <> 0_ (n,L) by POLYNOM7:def_1;
then  Support (f *' g) <> {} by POLYNOM7:1;
then A4: HT ((f *' g),T) in Support (f *' g) by TERMORD:def_6;
 HT ((f *' g),T) = (HT (f,T)) + (HT (g,T)) by TERMORD:31;
then  f *' g reduces_to (f *' g) - ((((f *' g) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)),g, HT ((f *' g),T),T by A3, A1, A4, POLYRED:def_5;
then A5: ( g in {g} & f *' g reduces_to (f *' g) - ((((f *' g) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)),g,T ) by POLYRED:def_6, TARSKI:def_1;
(f *' g) - ((((f *' g) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)) = (f *' g) - (((HC ((f *' g),T)) / (HC (g,T))) * ((HT (f,T)) *' g)) by TERMORD:def_7
.= (f *' g) - ((((HC (f,T)) * (HC (g,T))) / (HC (g,T))) * ((HT (f,T)) *' g)) by TERMORD:32
.= (f *' g) - ((((HC (f,T)) * (HC (g,T))) * ((HC (g,T)) ")) * ((HT (f,T)) *' g)) by VECTSP_1:def_11
.= (f *' g) - (((HC (f,T)) * ((HC (g,T)) * ((HC (g,T)) "))) * ((HT (f,T)) *' g)) by GROUP_1:def_3
.= (f *' g) - (((HC (f,T)) * (1. L)) * ((HT (f,T)) *' g)) by A2, VECTSP_1:def_10
.= (f *' g) - ((HC (f,T)) * ((HT (f,T)) *' g)) by VECTSP_1:def_6
.= (f *' g) - ((Monom ((HC (f,T)),(HT (f,T)))) *' g) by POLYRED:22
.= (f *' g) - ((HM (f,T)) *' g) by TERMORD:def_8
.= (f *' g) + (- ((HM (f,T)) *' g)) by POLYNOM1:def_6
.= (f *' g) + (r *' g) by POLYRED:6
.= g *' (f + (- (HM (f,T)))) by POLYNOM1:26
.= (f - (HM (f,T))) *' g by POLYNOM1:def_6
.= (Red (f,T)) *' g by TERMORD:def_9 ;
hence  f *' g reduces_to (Red (f,T)) *' g,{g},T by A5, POLYRED:def_7; ::_thesis: verum
end;

theorem :: GROEB_3:48
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L
for p being Polynomial of n,L st p . (HT ((f *' g),T)) = 0. L holds
(f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L
for p being Polynomial of n,L st p . (HT ((f *' g),T)) = 0. L holds
(f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L
for p being Polynomial of n,L st p . (HT ((f *' g),T)) = 0. L holds
(f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being non-zero Polynomial of n,L
for p being Polynomial of n,L st p . (HT ((f *' g),T)) = 0. L holds
(f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
let f, g be non-zero Polynomial of n,L; ::_thesis: for p being Polynomial of n,L st p . (HT ((f *' g),T)) = 0. L holds
(f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
let p be Polynomial of n,L; ::_thesis: ( p . (HT ((f *' g),T)) = 0. L implies (f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T )
assume A1: p . (HT ((f *' g),T)) = 0. L ; ::_thesis: (f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T
 f *' g <> 0_ (n,L) by POLYNOM7:def_1;
then  Support (f *' g) <> {} by POLYNOM7:1;
then  HT ((f *' g),T) in Support (f *' g) by TERMORD:def_6;
then A2: (f *' g) . (HT ((f *' g),T)) <> 0. L by POLYNOM1:def_3;
reconsider r = - (HM (f,T)) as Polynomial of n,L ;
set fg = (f *' g) + p;
set q = ((f *' g) + p) - (((((f *' g) + p) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g));
A3: HT ((f *' g),T) = (HT (f,T)) + (HT (g,T)) by TERMORD:31;
A4: g <> 0_ (n,L) by POLYNOM7:def_1;
A5: HC (g,T) <> 0. L ;
((f *' g) + p) . (HT ((f *' g),T)) = ((f *' g) . (HT ((f *' g),T))) + (p . (HT ((f *' g),T))) by POLYNOM1:15
.= (f *' g) . (HT ((f *' g),T)) by A1, RLVECT_1:def_4 ;
then A6: HT ((f *' g),T) in Support ((f *' g) + p) by A2, POLYNOM1:def_3;
then  (f *' g) + p <> 0_ (n,L) by POLYNOM7:1;
then  (f *' g) + p reduces_to ((f *' g) + p) - (((((f *' g) + p) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)),g, HT ((f *' g),T),T by A6, A4, A3, POLYRED:def_5;
then A7: ( g in {g} & (f *' g) + p reduces_to ((f *' g) + p) - (((((f *' g) + p) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)),g,T ) by POLYRED:def_6, TARSKI:def_1;
((f *' g) + p) - (((((f *' g) + p) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)) = ((f *' g) + p) - (((((f *' g) . (HT ((f *' g),T))) + (0. L)) / (HC (g,T))) * ((HT (f,T)) *' g)) by A1, POLYNOM1:15
.= ((f *' g) + p) - ((((f *' g) . (HT ((f *' g),T))) / (HC (g,T))) * ((HT (f,T)) *' g)) by RLVECT_1:def_4
.= ((f *' g) + p) - (((HC ((f *' g),T)) / (HC (g,T))) * ((HT (f,T)) *' g)) by TERMORD:def_7
.= ((f *' g) + p) - ((((HC (f,T)) * (HC (g,T))) / (HC (g,T))) * ((HT (f,T)) *' g)) by TERMORD:32
.= ((f *' g) + p) - ((((HC (f,T)) * (HC (g,T))) * ((HC (g,T)) ")) * ((HT (f,T)) *' g)) by VECTSP_1:def_11
.= ((f *' g) + p) - (((HC (f,T)) * ((HC (g,T)) * ((HC (g,T)) "))) * ((HT (f,T)) *' g)) by GROUP_1:def_3
.= ((f *' g) + p) - (((HC (f,T)) * (1. L)) * ((HT (f,T)) *' g)) by A5, VECTSP_1:def_10
.= ((f *' g) + p) - ((HC (f,T)) * ((HT (f,T)) *' g)) by VECTSP_1:def_6
.= ((f *' g) + p) - ((Monom ((HC (f,T)),(HT (f,T)))) *' g) by POLYRED:22
.= ((f *' g) + p) - ((HM (f,T)) *' g) by TERMORD:def_8
.= ((f *' g) + p) + (- ((HM (f,T)) *' g)) by POLYNOM1:def_6
.= ((f *' g) + p) + (r *' g) by POLYRED:6
.= ((f *' g) + (r *' g)) + p by POLYNOM1:21
.= (g *' (f + (- (HM (f,T))))) + p by POLYNOM1:26
.= ((f - (HM (f,T))) *' g) + p by POLYNOM1:def_6
.= ((Red (f,T)) *' g) + p by TERMORD:def_9 ;
hence  (f *' g) + p reduces_to ((Red (f,T)) *' g) + p,{g},T by A7, POLYRED:def_7; ::_thesis: verum
end;

theorem Th49: :: GROEB_3:49
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces - f, - g
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces - f, - g
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces - f, - g
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces - f, - g
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces - f, - g
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f,g implies PolyRedRel (P,T) reduces - f, - g )
assume  PolyRedRel (P,T) reduces f,g ; ::_thesis: PolyRedRel (P,T) reduces - f, - g
then consider R being RedSequence of PolyRedRel (P,T) such that
A1: R . 1 = f and
A2: R . (len R) = g by REWRITE1:def_3;
defpred S1[ Element of NAT ] means  for q being Polynomial of n,L st q = R . $1 holds
PolyRedRel (P,T) reduces - f, - q;
A3: 1 <= len R by NAT_1:14;
A4: now__::_thesis:_for_k_being_Element_of_NAT_st_1_<=_k_&_k_<_len_R_&_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len R & S1[k] implies S1[k + 1] )
assume A5: ( 1 <= k & k < len R ) ; ::_thesis: ( S1[k] implies S1[k + 1] )
then  1 < len R by XXREAL_0:2;
then reconsider p = R . k as Polynomial of n,L by A5, Lm4;
assume  S1[k] ; ::_thesis: S1[k + 1]
then A6: PolyRedRel (P,T) reduces - f, - p ;
now__::_thesis:_for_q_being_Polynomial_of_n,L_st_q_=_R_._(k_+_1)_holds_
PolyRedRel_(P,T)_reduces_-_f,_-_q
let q be Polynomial of n,L; ::_thesis: ( q = R . (k + 1) implies PolyRedRel (P,T) reduces - f, - q )
assume A7: q = R . (k + 1) ; ::_thesis: PolyRedRel (P,T) reduces - f, - q
 ( 1 <= k + 1 & k + 1 <= len R ) by A5, NAT_1:13;
then A8: k + 1 in dom R by FINSEQ_3:25;
 k in dom R by A5, FINSEQ_3:25;
then  [(R . k),(R . (k + 1))] in PolyRedRel (P,T) by A8, REWRITE1:def_2;
then  p reduces_to q,P,T by A7, POLYRED:def_13;
then consider l being Polynomial of n,L such that
A9: l in P and
A10: p reduces_to q,l,T by POLYRED:def_7;
 - p reduces_to - q,l,T by A10, Th45;
then  - p reduces_to - q,P,T by A9, POLYRED:def_7;
then  [(- p),(- q)] in PolyRedRel (P,T) by POLYRED:def_13;
then  PolyRedRel (P,T) reduces - p, - q by REWRITE1:15;
hence  PolyRedRel (P,T) reduces - f, - q by A6, REWRITE1:16; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A11: S1[1] by A1, REWRITE1:12;
 for i being Element of NAT st 1 <= i & i <= len R holds
S1[i] from INT_1:sch_7(A11, A4);
hence  PolyRedRel (P,T) reduces - f, - g by A2, A3; ::_thesis: verum
end;

theorem :: GROEB_3:50
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st PolyRedRel ({p},T) reduces f,f1 & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
PolyRedRel ({p},T) reduces f + g,f1 + g
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st PolyRedRel ({p},T) reduces f,f1 & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
PolyRedRel ({p},T) reduces f + g,f1 + g
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, f1, g, p being Polynomial of n,L st PolyRedRel ({p},T) reduces f,f1 & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
PolyRedRel ({p},T) reduces f + g,f1 + g
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, f1, g, p being Polynomial of n,L st PolyRedRel ({p},T) reduces f,f1 & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) holds
PolyRedRel ({p},T) reduces f + g,f1 + g
let f, f1, g, p be Polynomial of n,L; ::_thesis: ( PolyRedRel ({p},T) reduces f,f1 & ( for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ) implies PolyRedRel ({p},T) reduces f + g,f1 + g )
assume that
A1: PolyRedRel ({p},T) reduces f,f1 and
A2: for b1 being bag of n st b1 in Support g holds
not HT (p,T) divides b1 ; ::_thesis: PolyRedRel ({p},T) reduces f + g,f1 + g
consider R being RedSequence of PolyRedRel ({p},T) such that
A3: R . 1 = f and
A4: R . (len R) = f1 by A1, REWRITE1:def_3;
defpred S1[ Element of NAT ] means  for q being Polynomial of n,L st q = R . $1 holds
PolyRedRel ({p},T) reduces f + g,q + g;
A5: now__::_thesis:_for_k_being_Element_of_NAT_st_1_<=_k_&_k_<_len_R_&_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len R & S1[k] implies S1[k + 1] )
assume A6: ( 1 <= k & k < len R ) ; ::_thesis: ( S1[k] implies S1[k + 1] )
then  1 < len R by XXREAL_0:2;
then reconsider h = R . k as Polynomial of n,L by A6, Lm4;
assume  S1[k] ; ::_thesis: S1[k + 1]
then A7: PolyRedRel ({p},T) reduces f + g,h + g ;
now__::_thesis:_for_q_being_Polynomial_of_n,L_st_q_=_R_._(k_+_1)_holds_
PolyRedRel_({p},T)_reduces_f_+_g,q_+_g
let q be Polynomial of n,L; ::_thesis: ( q = R . (k + 1) implies PolyRedRel ({p},T) reduces f + g,q + g )
assume A8: q = R . (k + 1) ; ::_thesis: PolyRedRel ({p},T) reduces f + g,q + g
 ( 1 <= k + 1 & k + 1 <= len R ) by A6, NAT_1:13;
then A9: k + 1 in dom R by FINSEQ_3:25;
 k in dom R by A6, FINSEQ_3:25;
then  [(R . k),(R . (k + 1))] in PolyRedRel ({p},T) by A9, REWRITE1:def_2;
then  h reduces_to q,{p},T by A8, POLYRED:def_13;
then  h + g reduces_to q + g,{p},T by A2, Th46;
then  [(h + g),(q + g)] in PolyRedRel ({p},T) by POLYRED:def_13;
then  PolyRedRel ({p},T) reduces h + g,q + g by REWRITE1:15;
hence  PolyRedRel ({p},T) reduces f + g,q + g by A7, REWRITE1:16; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A10: 1 <= len R by NAT_1:14;
A11: S1[1] by A3, REWRITE1:12;
 for i being Element of NAT st 1 <= i & i <= len R holds
S1[i] from INT_1:sch_7(A11, A5);
hence  PolyRedRel ({p},T) reduces f + g,f1 + g by A4, A10; ::_thesis: verum
end;

theorem Th51: :: GROEB_3:51
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, g being non-zero Polynomial of n,L holds PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being non-zero Polynomial of n,L holds PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
let f, g be non-zero Polynomial of n,L; ::_thesis: PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
defpred S1[ Element of NAT ] means  for f being Polynomial of n,L st card (Support f) = $1 holds
PolyRedRel ({g},T) reduces f *' g, 0_ (n,L);
A1: ex n being Element of NAT st card (Support f) = n ;
A2: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f_being_Polynomial_of_n,L_st_card_(Support_f)_=_k_+_1_holds_
PolyRedRel_({g},T)_reduces_f_*'_g,_0__(n,L)
let f be Polynomial of n,L; ::_thesis: ( card (Support f) = k + 1 implies PolyRedRel ({g},T) reduces f *' g, 0_ (n,L) )
set rf = Red (f,T);
assume A4: card (Support f) = k + 1 ; ::_thesis: PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
now__::_thesis:_not_f_=_0__(n,L)
assume  f = 0_ (n,L) ; ::_thesis: contradiction
then  Support f = {} by POLYNOM7:1;
hence  contradiction by A4; ::_thesis: verum
end;
then reconsider f1 = f as non-zero Polynomial of n,L by POLYNOM7:def_1;
 f1 *' g reduces_to (Red (f,T)) *' g,{g},T by Th47;
then  [(f1 *' g),((Red (f,T)) *' g)] in PolyRedRel ({g},T) by POLYRED:def_13;
then A5: PolyRedRel ({g},T) reduces f *' g,(Red (f,T)) *' g by REWRITE1:15;
 f1 <> 0_ (n,L) by POLYNOM7:def_1;
then  Support f <> {} by POLYNOM7:1;
then  HT (f,T) in Support f by TERMORD:def_6;
then  for u being set st u in {(HT (f,T))} holds
u in Support f by TARSKI:def_1;
then A6: {(HT (f,T))} c= Support f by TARSKI:def_3;
 Support (Red (f,T)) = (Support f) \ {(HT (f,T))} by TERMORD:36;
then  card (Support (Red (f,T))) = (card (Support f)) - (card {(HT (f,T))}) by A6, CARD_2:44
.= (k + 1) - 1 by A4, CARD_1:30
.= k + 0 ;
then  PolyRedRel ({g},T) reduces (Red (f,T)) *' g, 0_ (n,L) by A3;
hence  PolyRedRel ({g},T) reduces f *' g, 0_ (n,L) by A5, REWRITE1:16; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
now__::_thesis:_for_f_being_Polynomial_of_n,L_st_card_(Support_f)_=_0_holds_
PolyRedRel_({g},T)_reduces_f_*'_g,_0__(n,L)
let f be Polynomial of n,L; ::_thesis: ( card (Support f) = 0 implies PolyRedRel ({g},T) reduces f *' g, 0_ (n,L) )
assume  card (Support f) = 0 ; ::_thesis: PolyRedRel ({g},T) reduces f *' g, 0_ (n,L)
then  Support f = {} ;
then  f = 0_ (n,L) by POLYNOM7:1;
then  f *' g = 0_ (n,L) by POLYRED:5;
hence  PolyRedRel ({g},T) reduces f *' g, 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
then A7: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A2);
hence  PolyRedRel ({g},T) reduces f *' g, 0_ (n,L) by A1; ::_thesis: verum
end;

begin

theorem Th52: :: GROEB_3:52
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
let p1, p2 be Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint implies for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 )
assume A1: HT (p1,T), HT (p2,T) are_disjoint ; ::_thesis: for b1, b2 being bag of n st b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) holds
not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
A2: ( Support (Red (p1,T)) c= Support p1 & Support (Red (p2,T)) c= Support p2 ) by TERMORD:35;
let b1, b2 be bag of n; ::_thesis: ( b1 in Support (Red (p1,T)) & b2 in Support (Red (p2,T)) implies not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 )
assume that
A3: b1 in Support (Red (p1,T)) and
A4: b2 in Support (Red (p2,T)) ; ::_thesis: not (HT (p1,T)) + b2 = (HT (p2,T)) + b1
now__::_thesis:_not_b1_=_HT_(p1,T)
assume  b1 = HT (p1,T) ; ::_thesis: contradiction
then  (Red (p1,T)) . b1 = 0. L by TERMORD:39;
hence  contradiction by A3, POLYNOM1:def_3; ::_thesis: verum
end;
hence  not (HT (p1,T)) + b2 = (HT (p2,T)) + b1 by A1, A3, A4, A2, Lm5; ::_thesis: verum
end;

theorem Th53: :: GROEB_3:53
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))
let p1, p2 be Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint implies S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))) )
assume  HT (p1,T), HT (p2,T) are_disjoint ; ::_thesis: S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))
then  lcm ((HT (p1,T)),(HT (p2,T))) = (HT (p1,T)) + (HT (p2,T)) by GROEB_2:5;
hence  S-Poly (p1,p2,T) = ((HC (p2,T)) * ((((HT (p1,T)) + (HT (p2,T))) / (HT (p1,T))) *' p1)) - ((HC (p1,T)) * ((((HT (p1,T)) + (HT (p2,T))) / (HT (p2,T))) *' p2)) by GROEB_2:def_4
.= ((HC (p2,T)) * ((HT (p2,T)) *' p1)) - ((HC (p1,T)) * ((((HT (p1,T)) + (HT (p2,T))) / (HT (p2,T))) *' p2)) by Th1
.= ((HC (p2,T)) * ((HT (p2,T)) *' p1)) - ((HC (p1,T)) * ((HT (p1,T)) *' p2)) by Th1
.= ((HC (p2,T)) * ((HT (p2,T)) *' ((HM (p1,T)) + (Red (p1,T))))) - ((HC (p1,T)) * ((HT (p1,T)) *' p2)) by TERMORD:38
.= ((HC (p2,T)) * ((HT (p2,T)) *' ((HM (p1,T)) + (Red (p1,T))))) - ((HC (p1,T)) * ((HT (p1,T)) *' ((HM (p2,T)) + (Red (p2,T))))) by TERMORD:38
.= ((Monom ((HC (p2,T)),(HT (p2,T)))) *' ((HM (p1,T)) + (Red (p1,T)))) - ((HC (p1,T)) * ((HT (p1,T)) *' ((HM (p2,T)) + (Red (p2,T))))) by POLYRED:22
.= ((Monom ((HC (p2,T)),(HT (p2,T)))) *' ((HM (p1,T)) + (Red (p1,T)))) - ((Monom ((HC (p1,T)),(HT (p1,T)))) *' ((HM (p2,T)) + (Red (p2,T)))) by POLYRED:22
.= ((HM (p2,T)) *' ((HM (p1,T)) + (Red (p1,T)))) - ((Monom ((HC (p1,T)),(HT (p1,T)))) *' ((HM (p2,T)) + (Red (p2,T)))) by TERMORD:def_8
.= ((HM (p2,T)) *' ((HM (p1,T)) + (Red (p1,T)))) - ((HM (p1,T)) *' ((HM (p2,T)) + (Red (p2,T)))) by TERMORD:def_8
.= (((HM (p2,T)) *' (HM (p1,T))) + ((HM (p2,T)) *' (Red (p1,T)))) - ((HM (p1,T)) *' ((HM (p2,T)) + (Red (p2,T)))) by POLYNOM1:26
.= (((HM (p2,T)) *' (HM (p1,T))) + ((HM (p2,T)) *' (Red (p1,T)))) - (((HM (p1,T)) *' (HM (p2,T))) + ((HM (p1,T)) *' (Red (p2,T)))) by POLYNOM1:26
.= (((HM (p2,T)) *' (HM (p1,T))) + ((HM (p2,T)) *' (Red (p1,T)))) + (- (((HM (p1,T)) *' (HM (p2,T))) + ((HM (p1,T)) *' (Red (p2,T))))) by POLYNOM1:def_6
.= (((HM (p2,T)) *' (HM (p1,T))) + ((HM (p2,T)) *' (Red (p1,T)))) + ((- ((HM (p1,T)) *' (HM (p2,T)))) + (- ((HM (p1,T)) *' (Red (p2,T))))) by POLYRED:1
.= ((HM (p2,T)) *' (Red (p1,T))) + (((HM (p2,T)) *' (HM (p1,T))) + ((- ((HM (p1,T)) *' (HM (p2,T)))) + (- ((HM (p1,T)) *' (Red (p2,T)))))) by POLYNOM1:21
.= ((HM (p2,T)) *' (Red (p1,T))) + ((((HM (p2,T)) *' (HM (p1,T))) + (- ((HM (p1,T)) *' (HM (p2,T))))) + (- ((HM (p1,T)) *' (Red (p2,T))))) by POLYNOM1:21
.= ((HM (p2,T)) *' (Red (p1,T))) + ((0_ (n,L)) + (- ((HM (p1,T)) *' (Red (p2,T))))) by POLYRED:3
.= ((HM (p2,T)) *' (Red (p1,T))) + (- ((HM (p1,T)) *' (Red (p2,T)))) by POLYRED:2
.= ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))) by POLYNOM1:def_6 ;
::_thesis: verum
end;

theorem Th54: :: GROEB_3:54
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1)
let p1, p2 be Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint implies S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1) )
reconsider r1 = - (Red (p1,T)), r2 = - (Red (p2,T)) as Polynomial of n,L ;
r2 *' (Red (p1,T)) = - ((Red (p2,T)) *' (Red (p1,T))) by POLYRED:6
.= r1 *' (Red (p2,T)) by POLYRED:6 ;
then A1: (r2 *' (Red (p1,T))) + (- (r1 *' (Red (p2,T)))) = 0_ (n,L) by POLYRED:3;
assume  HT (p1,T), HT (p2,T) are_disjoint ; ::_thesis: S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1)
hence  S-Poly (p1,p2,T) = ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))) by Th53
.= ((p2 - (Red (p2,T))) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))) by Th15
.= ((p2 - (Red (p2,T))) *' (Red (p1,T))) - ((p1 - (Red (p1,T))) *' (Red (p2,T))) by Th15
.= ((p2 + (- (Red (p2,T)))) *' (Red (p1,T))) - ((p1 - (Red (p1,T))) *' (Red (p2,T))) by POLYNOM1:def_6
.= ((p2 + (- (Red (p2,T)))) *' (Red (p1,T))) - ((p1 + (- (Red (p1,T)))) *' (Red (p2,T))) by POLYNOM1:def_6
.= ((p2 *' (Red (p1,T))) + (r2 *' (Red (p1,T)))) - ((p1 + (- (Red (p1,T)))) *' (Red (p2,T))) by POLYNOM1:26
.= ((p2 *' (Red (p1,T))) + (r2 *' (Red (p1,T)))) - ((p1 *' (Red (p2,T))) + (r1 *' (Red (p2,T)))) by POLYNOM1:26
.= ((p2 *' (Red (p1,T))) + (r2 *' (Red (p1,T)))) + (- ((p1 *' (Red (p2,T))) + (r1 *' (Red (p2,T))))) by POLYNOM1:def_6
.= ((p2 *' (Red (p1,T))) + (r2 *' (Red (p1,T)))) + ((- (p1 *' (Red (p2,T)))) + (- (r1 *' (Red (p2,T))))) by POLYRED:1
.= (p2 *' (Red (p1,T))) + ((r2 *' (Red (p1,T))) + ((- (r1 *' (Red (p2,T)))) + (- (p1 *' (Red (p2,T)))))) by POLYNOM1:21
.= (p2 *' (Red (p1,T))) + ((0_ (n,L)) + (- (p1 *' (Red (p2,T))))) by A1, POLYNOM1:21
.= (p2 *' (Red (p1,T))) + (- (p1 *' (Red (p2,T)))) by POLYRED:2
.= ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1) by POLYNOM1:def_6 ;
::_thesis: verum
end;

theorem Th55: :: GROEB_3:55
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being non-zero Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint & Red (p1,T) is non-zero & Red (p2,T) is non-zero holds
PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T))
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being non-zero Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint & Red (p1,T) is non-zero & Red (p2,T) is non-zero holds
PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T))
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being non-zero Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint & Red (p1,T) is non-zero & Red (p2,T) is non-zero holds
PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T))
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being non-zero Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint & Red (p1,T) is non-zero & Red (p2,T) is non-zero holds
PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T))
let p1, p2 be non-zero Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint & Red (p1,T) is non-zero & Red (p2,T) is non-zero implies PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T)) )
assume that
A1: HT (p1,T), HT (p2,T) are_disjoint and
A2: ( Red (p1,T) is non-zero & Red (p2,T) is non-zero ) ; ::_thesis: PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T))
reconsider red1 = Red (p1,T), red2 = Red (p2,T) as non-zero Polynomial of n,L by A2;
set j = card (Support p2);
defpred S1[ Element of NAT ] means  for m being Element of NAT st m <= card (Support p2) & card (Support (Low (p2,T,m))) = $1 holds
PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m)));
now__::_thesis:_not_card_(Support_p2)_=_0
assume  card (Support p2) = 0 ; ::_thesis: contradiction
then  Support p2 = {} ;
then  p2 = 0_ (n,L) by POLYNOM7:1;
hence  contradiction by POLYNOM7:def_1; ::_thesis: verum
end;
then A3: 1 <= card (Support p2) by NAT_1:14;
then  1 - 1 <= (card (Support p2)) - 1 by XREAL_1:9;
then reconsider j9 = (card (Support p2)) - 1 as Element of NAT by INT_1:3;
A4: (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,1))))) + ((Red (p1,T)) *' (Low (p2,T,1))) = (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Red (p2,T))))) + ((Red (p1,T)) *' (Low (p2,T,1))) by Th36
.= (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (0_ (n,L)))) + ((Red (p1,T)) *' (Low (p2,T,1))) by POLYNOM1:24
.= (((HM (p2,T)) *' (Red (p1,T))) - (0_ (n,L))) + ((Red (p1,T)) *' (Low (p2,T,1))) by POLYRED:5
.= ((HM (p2,T)) *' (Red (p1,T))) + ((Red (p1,T)) *' (Low (p2,T,1))) by POLYRED:4
.= ((HM (p2,T)) *' (Red (p1,T))) + ((Red (p1,T)) *' (Red (p2,T))) by Th36
.= ((HM (p2,T)) + (Red (p2,T))) *' (Red (p1,T)) by POLYNOM1:26
.= p2 *' (Red (p1,T)) by TERMORD:38 ;
 p2 <> 0_ (n,L) by POLYNOM7:def_1;
then  Support p2 <> {} by POLYNOM7:1;
then  HT (p2,T) in Support p2 by TERMORD:def_6;
then  for t being set st t in {(HT (p2,T))} holds
t in Support p2 by TARSKI:def_1;
then A5: {(HT (p2,T))} c= Support p2 by TARSKI:def_3;
A6: card (Support red2) = card ((Support p2) \ {(HT (p2,T))}) by TERMORD:36
.= (card (Support p2)) - (card {(HT (p2,T))}) by A5, CARD_2:44
.= (card (Support p2)) - 1 by CARD_2:42 ;
then A7: card (Support (Low (p2,T,1))) = j9 by Th36;
A8: for k being Element of NAT st 0 <= k & k < j9 & S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( 0 <= k & k < j9 & S1[k] implies S1[k + 1] )
assume that
 0 <= k and
A9: k < j9 ; ::_thesis: ( not S1[k] or S1[k + 1] )
now__::_thesis:_(_S1[k]_implies_S1[k_+_1]_)
assume A10: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_m_being_Element_of_NAT_st_m_<=_card_(Support_p2)_&_card_(Support_(Low_(p2,T,m)))_=_k_+_1_holds_
PolyRedRel_({p1},T)_reduces_((HM_(p2,T))_*'_(Red_(p1,T)))_-_((HM_(p1,T))_*'_(Red_(p2,T))),(((HM_(p2,T))_*'_(Red_(p1,T)))_-_((HM_(p1,T))_*'_((Red_(p2,T))_-_(Low_(p2,T,m)))))_+_((Red_(p1,T))_*'_(Low_(p2,T,m)))
 ( HT (((HM (p2,T)) *' red1),T) = (HT ((HM (p2,T)),T)) + (HT (red1,T)) & HC (((HM (p2,T)) *' red1),T) <> 0. L ) by TERMORD:31;
then A11: ((HM (p2,T)) *' red1) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) <> 0. L by TERMORD:def_7;
A12: Support (Red (p2,T)) c= Support p2 by TERMORD:35;
 red2 <> 0_ (n,L) by POLYNOM7:def_1;
then A13: Support red2 <> {} by POLYNOM7:1;
let m be Element of NAT ; ::_thesis: ( m <= card (Support p2) & card (Support (Low (p2,T,m))) = k + 1 implies PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) )
assume that
A14: m <= card (Support p2) and
A15: card (Support (Low (p2,T,m))) = k + 1 ; ::_thesis: PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m)))
set m9 = m + 1;
now__::_thesis:_not_m_=_card_(Support_p2)
assume  m = card (Support p2) ; ::_thesis: contradiction
then  Low (p2,T,m) = 0_ (n,L) by Th35;
hence  contradiction by A15, CARD_1:27, POLYNOM7:1; ::_thesis: verum
end;
then A16: m < card (Support p2) by A14, XXREAL_0:1;
then  card ((Support (Low (p2,T,m))) \ (Support (Low (p2,T,(m + 1))))) = (card (Support (Low (p2,T,m)))) - (card (Support (Low (p2,T,(m + 1))))) by Th41, CARD_2:44;
then A17: (k + 1) - (card (Support (Low (p2,T,(m + 1))))) = card {(HT ((Low (p2,T,m)),T))} by A15, A16, Th42
.= 1 by CARD_1:30 ;
set f = (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))));
A18: (HT ((HM (p2,T)),T)) + (HT (red1,T)) is Element of Bags n by PRE_POLY:def_12;
A19: m + 1 <= card (Support p2) by A16, NAT_1:13;
now__::_thesis:_not_(HT_((HM_(p2,T)),T))_+_(HT_(red1,T))_in_Support_(red1_*'_(Low_(p2,T,(m_+_1))))
A20: Support (red1 *' (Low (p2,T,(m + 1)))) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support red1 & t in Support (Low (p2,T,(m + 1))) )  }  by TERMORD:30;
assume  (HT ((HM (p2,T)),T)) + (HT (red1,T)) in Support (red1 *' (Low (p2,T,(m + 1)))) ; ::_thesis: contradiction
then  (HT ((HM (p2,T)),T)) + (HT (red1,T)) in  {  (s + t) where s, t is Element of Bags n : ( s in Support red1 & t in Support (Low (p2,T,(m + 1))) )  }  by A20;
then consider s, t being Element of Bags n such that
A21: (HT ((HM (p2,T)),T)) + (HT (red1,T)) = s + t and
A22: s in Support red1 and
A23: t in Support (Low (p2,T,(m + 1))) ;
A24: t < HT (p2,T),T
proof
now__::_thesis:_(_(_m_+_1_=_card_(Support_p2)_&_contradiction_)_or_(_m_+_1_<>_card_(Support_p2)_&_t_<_HT_(p2,T),T_)_)
percases ( m + 1 = card (Support p2) or m + 1 <> card (Support p2) ) ;
case m + 1 = card (Support p2) ; ::_thesis: contradiction
then  Low (p2,T,(m + 1)) = 0_ (n,L) by Th35;
hence  contradiction by A23, POLYNOM7:1; ::_thesis: verum
end;
caseA25: m + 1 <> card (Support p2) ; ::_thesis: t < HT (p2,T),T
A26: t <= HT ((Low (p2,T,(m + 1))),T),T by A23, TERMORD:def_6;
 m + 1 < card (Support p2) by A19, A25, XXREAL_0:1;
hence  t < HT (p2,T),T by A26, Th3, Th39; ::_thesis: verum
end;
end;
end;
hence  t < HT (p2,T),T ; ::_thesis: verum
end;
 s <= HT (red1,T),T by A22, TERMORD:def_6;
then  s + t < (HT (p2,T)) + (HT (red1,T)),T by A24, Th5;
then  s + t < (HT ((HM (p2,T)),T)) + (HT (red1,T)),T by TERMORD:26;
hence  contradiction by A21, TERMORD:def_3; ::_thesis: verum
end;
then A27: (red1 *' (Low (p2,T,(m + 1)))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) = 0. L by A18, POLYNOM1:def_3;
A28: 1 <= m + 1 by NAT_1:14;
now__::_thesis:_not_(HT_((HM_(p2,T)),T))_+_(HT_(red1,T))_in_Support_((HM_(p1,T))_*'_(red2_-_(Low_(p2,T,(m_+_1)))))
 red1 <> 0_ (n,L) by POLYNOM7:def_1;
then  Support red1 <> {} by POLYNOM7:1;
then A29: ( HT ((HM (p2,T)),T) = HT (p2,T) & HT (red1,T) in Support red1 ) by TERMORD:26, TERMORD:def_6;
A30: Support ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) c=  {  (s + t) where s, t is Element of Bags n : ( s in Support (HM (p1,T)) & t in Support (red2 - (Low (p2,T,(m + 1)))) )  }  by TERMORD:30;
assume  (HT ((HM (p2,T)),T)) + (HT (red1,T)) in Support ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) ; ::_thesis: contradiction
then A31: (HT ((HM (p2,T)),T)) + (HT (red1,T)) in  {  (s + t) where s, t is Element of Bags n : ( s in Support (HM (p1,T)) & t in Support (red2 - (Low (p2,T,(m + 1)))) )  }  by A30;
 ( red2 - (Low (p2,T,(m + 1))) = red2 + (- (Low (p2,T,(m + 1)))) & Support (- (Low (p2,T,(m + 1)))) = Support (Low (p2,T,(m + 1))) ) by GROEB_1:5, POLYNOM1:def_6;
then A32: Support (red2 - (Low (p2,T,(m + 1)))) c= (Support red2) \/ (Support (Low (p2,T,(m + 1)))) by POLYNOM1:20;
consider s, t being Element of Bags n such that
A33: (HT ((HM (p2,T)),T)) + (HT (red1,T)) = s + t and
A34: s in Support (HM (p1,T)) and
A35: t in Support (red2 - (Low (p2,T,(m + 1)))) by A31;
A36: Support (Low (p2,T,(m + 1))) c= Support red2 by A19, A28, Th27;
A37: t in Support red2
proof
now__::_thesis:_(_(_t_in_Support_red2_&_t_in_Support_red2_)_or_(_t_in_Support_(Low_(p2,T,(m_+_1)))_&_t_in_Support_red2_)_)
percases ( t in Support red2 or t in Support (Low (p2,T,(m + 1))) ) by A35, A32, XBOOLE_0:def_3;
case t in Support red2 ; ::_thesis: t in Support red2
hence  t in Support red2 ; ::_thesis: verum
end;
case t in Support (Low (p2,T,(m + 1))) ; ::_thesis: t in Support red2
hence  t in Support red2 by A36; ::_thesis: verum
end;
end;
end;
hence  t in Support red2 ; ::_thesis: verum
end;
 HM (p1,T) <> 0_ (n,L) by POLYNOM7:def_1;
then  Support (HM (p1,T)) <> {} by POLYNOM7:1;
then  Support (HM (p1,T)) = {(HT (p1,T))} by TERMORD:21;
then  s = HT (p1,T) by A34, TARSKI:def_1;
hence  contradiction by A1, A33, A29, A37, Th52; ::_thesis: verum
end;
then  ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) = 0. L by A18, POLYNOM1:def_3;
then A38: - (((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T)))) = 0. L by RLVECT_1:12;
A39: Support (Low (p2,T,(m + 1))) = Lower_Support (p2,T,(m + 1)) by A19, Lm3;
now__::_thesis:_not_HT_(red2,T)_in_Support_(Low_(p2,T,(m_+_1)))
assume A40: HT (red2,T) in Support (Low (p2,T,(m + 1))) ; ::_thesis: contradiction
A41: now__::_thesis:_for_t_being_set_st_t_in_Support_red2_holds_
t_in_Support_(Low_(p2,T,(m_+_1)))
let t be set ; ::_thesis: ( t in Support red2 implies t in Support (Low (p2,T,(m + 1))) )
assume A42: t in Support red2 ; ::_thesis: t in Support (Low (p2,T,(m + 1)))
then reconsider t9 = t as bag of n ;
 ( Support red2 c= Support p2 & t9 <= HT (red2,T),T ) by A42, TERMORD:35, TERMORD:def_6;
hence  t in Support (Low (p2,T,(m + 1))) by A19, A39, A40, A42, Th24; ::_thesis: verum
end;
 Support (Low (p2,T,(m + 1))) c= Support red2 by A19, A28, Th27;
then  for t being set st t in Support (Low (p2,T,(m + 1))) holds
t in Support red2 ;
hence  contradiction by A6, A9, A17, A41, TARSKI:1; ::_thesis: verum
end;
then  Low (p2,T,(m + 1)) <> red2 by A13, TERMORD:def_6;
then  (Red (p2,T)) - (Low (p2,T,(m + 1))) <> 0_ (n,L) by Th12;
then reconsider z1 = (Red (p2,T)) - (Low (p2,T,(m + 1))) as non-zero Polynomial of n,L by POLYNOM7:def_1;
reconsider z = (HM (p1,T)) *' z1 as non-zero Polynomial of n,L ;
 z1 = (Red (p2,T)) + (- (Low (p2,T,(m + 1)))) by POLYNOM1:def_6;
then  Support z1 c= (Support (Red (p2,T))) \/ (Support (- (Low (p2,T,(m + 1))))) by POLYNOM1:20;
then A43: Support z1 c= (Support (Red (p2,T))) \/ (Support (Low (p2,T,(m + 1)))) by GROEB_1:5;
 z <> 0_ (n,L) by POLYNOM7:def_1;
then  Support z <> {} by POLYNOM7:1;
then reconsider w = (card (Support z)) - 1 as Element of NAT by INT_1:5, NAT_1:14;
reconsider lowzw = Low (z,T,w) as non-zero Monomial of n,L by Th37;
set b = term lowzw;
set s = (term lowzw) / (HT (p1,T));
A44: Support ((HM (p1,T)) *' z1) c=  {  (t9 + t) where t9, t is Element of Bags n : ( t9 in Support (HM (p1,T)) & t in Support z1 )  }  by TERMORD:30;
 card (Support z) = w + 1 ;
then A45: w < card (Support z) by NAT_1:16;
then A46: Support lowzw c= Support z by Th26;
 lowzw <> 0_ (n,L) by POLYNOM7:def_1;
then  Support lowzw <> {} by POLYNOM7:1;
then  Support lowzw = {(term lowzw)} by POLYNOM7:7;
then A47: term lowzw in Support lowzw by TARSKI:def_1;
then  term lowzw in Support ((HM (p1,T)) *' z1) by A46;
then  term lowzw in  {  (t9 + t) where t9, t is Element of Bags n : ( t9 in Support (HM (p1,T)) & t in Support z1 )  }  by A44;
then consider t9, t being Element of Bags n such that
A48: term lowzw = t9 + t and
A49: t9 in Support (HM (p1,T)) and
A50: t in Support z1 ;
 HM (p1,T) <> 0_ (n,L) by POLYNOM7:def_1;
then  Support (HM (p1,T)) <> {} by POLYNOM7:1;
then  Support (HM (p1,T)) = {(term (HM (p1,T)))} by POLYNOM7:7
.= {(HT (p1,T))} by TERMORD:22 ;
then A51: t9 = HT (p1,T) by A49, TARSKI:def_1;
then A52: HT (p1,T) divides term lowzw by A48, PRE_POLY:50;
then A53: ((term lowzw) / (HT (p1,T))) + (HT (p1,T)) = term lowzw by GROEB_2:def_1;
A54: (term lowzw) / (HT (p1,T)) = (((term lowzw) / (HT (p1,T))) + (HT (p1,T))) -' (HT (p1,T)) by PRE_POLY:48
.= t by A48, A51, A53, PRE_POLY:48 ;
 (Support (Red (p2,T))) \/ (Support (Low (p2,T,(m + 1)))) c= (Support (Red (p2,T))) \/ (Support (Red (p2,T))) by A19, A28, Th27, XBOOLE_1:9;
then A55: Support z1 c= Support red2 by A43, XBOOLE_1:1;
then A56: (term lowzw) / (HT (p1,T)) in Support (Red (p2,T)) by A50, A54;
then  (term lowzw) / (HT (p1,T)) in (Support p2) \ {(HT (p2,T))} by TERMORD:36;
then  not (term lowzw) / (HT (p1,T)) in {(HT (p2,T))} by XBOOLE_0:def_5;
then A57: (term lowzw) / (HT (p1,T)) <> HT (p2,T) by TARSKI:def_1;
then A58: (Red (p2,T)) . ((term lowzw) / (HT (p1,T))) = p2 . ((term lowzw) / (HT (p1,T))) by A56, A12, TERMORD:40;
A59: now__::_thesis:_not_(term_lowzw)_/_(HT_(p1,T))_in_Support_(Low_(p2,T,(m_+_1)))
assume  (term lowzw) / (HT (p1,T)) in Support (Low (p2,T,(m + 1))) ; ::_thesis: contradiction
then A60: p2 . ((term lowzw) / (HT (p1,T))) = (Low (p2,T,(m + 1))) . ((term lowzw) / (HT (p1,T))) by Th16;
((Red (p2,T)) - (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T))) = ((Red (p2,T)) + (- (Low (p2,T,(m + 1))))) . ((term lowzw) / (HT (p1,T))) by POLYNOM1:def_6
.= ((Red (p2,T)) . ((term lowzw) / (HT (p1,T)))) + ((- (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T)))) by POLYNOM1:15
.= ((Red (p2,T)) . ((term lowzw) / (HT (p1,T)))) + (- ((Low (p2,T,(m + 1))) . ((term lowzw) / (HT (p1,T))))) by POLYNOM1:17
.= 0. L by A58, A60, RLVECT_1:5 ;
hence  contradiction by A50, A54, POLYNOM1:def_3; ::_thesis: verum
end;
A61: term lowzw is Element of Bags n by PRE_POLY:def_12;
A62: now__::_thesis:_not_((Red_(p1,T))_*'_(Low_(p2,T,(m_+_1))))_._(term_lowzw)_<>_0._L
assume  ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) . (term lowzw) <> 0. L ; ::_thesis: contradiction
then A63: term lowzw in Support ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) by A61, POLYNOM1:def_3;
 Support ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) c=  {  (u + v) where u, v is Element of Bags n : ( u in Support (Red (p1,T)) & v in Support (Low (p2,T,(m + 1))) )  }  by TERMORD:30;
then  term lowzw in  {  (u + v) where u, v is Element of Bags n : ( u in Support (Red (p1,T)) & v in Support (Low (p2,T,(m + 1))) )  }  by A63;
then consider t9, t being Element of Bags n such that
A64: term lowzw = t9 + t and
A65: t9 in Support (Red (p1,T)) and
A66: t in Support (Low (p2,T,(m + 1))) ;
A67: ((term lowzw) / (HT (p1,T))) + (HT (p1,T)) = t9 + t by A52, A64, GROEB_2:def_1;
now__::_thesis:_not_(term_lowzw)_/_(HT_(p1,T))_<_t,T
assume  (term lowzw) / (HT (p1,T)) < t,T ; ::_thesis: contradiction
then A68: (term lowzw) / (HT (p1,T)) <= t,T by TERMORD:def_3;
 t in Lower_Support (p2,T,(m + 1)) by A19, A66, Lm3;
then  (term lowzw) / (HT (p1,T)) in Lower_Support (p2,T,(m + 1)) by A19, A56, A12, A68, Th24;
hence  contradiction by A19, A59, Lm3; ::_thesis: verum
end;
then A69: t <= (term lowzw) / (HT (p1,T)),T by TERMORD:5;
 Support (Red (p1,T)) = (Support p1) \ {(HT (p1,T))} by TERMORD:36;
then  not t9 in {(HT (p1,T))} by A65, XBOOLE_0:def_5;
then A70: t9 <> HT (p1,T) by TARSKI:def_1;
 Support (Red (p1,T)) c= Support p1 by TERMORD:35;
then  t9 <= HT (p1,T),T by A65, TERMORD:def_6;
then  t9 < HT (p1,T),T by A70, TERMORD:def_3;
then  t + t9 < ((term lowzw) / (HT (p1,T))) + (HT (p1,T)),T by A69, Th6;
hence  contradiction by A67, TERMORD:def_3; ::_thesis: verum
end;
A71: now__::_thesis:_not_term_lowzw_in_Support_((HM_(p2,T))_*'_(Red_(p1,T)))
 HM (p2,T) <> 0_ (n,L) by POLYNOM7:def_1;
then A72: Support (HM (p2,T)) <> {} by POLYNOM7:1;
then  HT ((HM (p2,T)),T) in Support (HM (p2,T)) by TERMORD:def_6;
then A73: HT (p2,T) in Support (HM (p2,T)) by TERMORD:26;
A74: Support ((HM (p2,T)) *' (Red (p1,T))) c=  {  (u + v) where u, v is Element of Bags n : ( u in Support (HM (p2,T)) & v in Support (Red (p1,T)) )  }  by TERMORD:30;
assume  term lowzw in Support ((HM (p2,T)) *' (Red (p1,T))) ; ::_thesis: contradiction
then  term lowzw in  {  (u + v) where u, v is Element of Bags n : ( u in Support (HM (p2,T)) & v in Support (Red (p1,T)) )  }  by A74;
then consider t9, t being Element of Bags n such that
A75: term lowzw = t9 + t and
A76: t9 in Support (HM (p2,T)) and
A77: t in Support (Red (p1,T)) ;
 ex x being bag of n st Support (HM (p2,T)) = {x} by A72, POLYNOM7:6;
then  Support (HM (p2,T)) = {(HT (p2,T))} by A73, TARSKI:def_1;
then  t9 = HT (p2,T) by A76, TARSKI:def_1;
hence  contradiction by A1, A55, A50, A53, A54, A75, A77, Th52; ::_thesis: verum
end;
set g = ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1));
A78: (HT ((HM (p2,T)),T)) + (HT (red1,T)) is Element of Bags n by PRE_POLY:def_12;
A79: ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw) = ((((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) + (red1 *' (Low (p2,T,(m + 1))))) . (term lowzw) by POLYNOM1:def_6
.= ((((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) . (term lowzw)) + (0. L) by A62, POLYNOM1:15
.= (((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) . (term lowzw) by RLVECT_1:def_4
.= (((HM (p2,T)) *' red1) . (term lowzw)) + ((- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1)))))) . (term lowzw)) by POLYNOM1:15
.= (0. L) + ((- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1)))))) . (term lowzw)) by A61, A71, POLYNOM1:def_3
.= (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1)))))) . (term lowzw) by RLVECT_1:def_4
.= - (((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) . (term lowzw)) by POLYNOM1:17 ;
 w = (card (Support z1)) - 1 by Th10;
then reconsider lowz1w = Low (z1,T,w) as non-zero Monomial of n,L by Th37;
 w + 1 = ((card (Support z1)) - 1) + 1 by Th10;
then A80: w <= card (Support z1) by NAT_1:13;
 lowz1w <> 0_ (n,L) by POLYNOM7:def_1;
then  Support lowz1w <> {} by POLYNOM7:1;
then A81: Support lowz1w = {(term lowz1w)} by POLYNOM7:7;
 card (Support z) = card (Support z1) by Th10;
then ((term lowzw) / (HT (p1,T))) + (HT (p1,T)) = term ((HM (p1,T)) *' lowz1w) by A45, A53, Th44
.= (term (HM (p1,T))) + (term lowz1w) by Th7
.= (HT (p1,T)) + (term lowz1w) by TERMORD:22 ;
then A82: (term lowzw) / (HT (p1,T)) = ((HT (p1,T)) + (term lowz1w)) -' (HT (p1,T)) by PRE_POLY:48
.= term lowz1w by PRE_POLY:48 ;
then  (term lowzw) / (HT (p1,T)) in Support lowz1w by A81, TARSKI:def_1;
then A83: (term lowzw) / (HT (p1,T)) in Lower_Support (z1,T,w) by A80, Lm3;
 Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))) = HM ((Low (p2,T,m)),T)
proof
A84: now__::_thesis:_for_t_being_bag_of_n_st_t_in_Support_z1_holds_
(term_lowzw)_/_(HT_(p1,T))_<=_t,T
let t be bag of n; ::_thesis: ( t in Support z1 implies (term lowzw) / (HT (p1,T)) <= t,T )
assume A85: t in Support z1 ; ::_thesis: (term lowzw) / (HT (p1,T)) <= t,T
now__::_thesis:_not_t_<_(term_lowzw)_/_(HT_(p1,T)),T
assume A86: t < (term lowzw) / (HT (p1,T)),T ; ::_thesis: contradiction
then  t <= (term lowzw) / (HT (p1,T)),T by TERMORD:def_3;
then  t in Lower_Support (z1,T,w) by A80, A83, A85, Th24;
then  t in Support lowz1w by A80, Lm3;
then  t = term lowz1w by A81, TARSKI:def_1;
hence  contradiction by A82, A86, TERMORD:def_3; ::_thesis: verum
end;
hence  (term lowzw) / (HT (p1,T)) <= t,T by TERMORD:5; ::_thesis: verum
end;
set r = HT ((Low (p2,T,m)),T);
 (Support (Low (p2,T,m))) \ (Support (Low (p2,T,(m + 1)))) = {(HT ((Low (p2,T,m)),T))} by A16, Th42;
then A87: HT ((Low (p2,T,m)),T) in (Support (Low (p2,T,m))) \ (Support (Low (p2,T,(m + 1)))) by TARSKI:def_1;
then A88: not HT ((Low (p2,T,m)),T) in Support (Low (p2,T,(m + 1))) by XBOOLE_0:def_5;
A89: ((Red (p2,T)) - (Low (p2,T,(m + 1)))) . (HT ((Low (p2,T,m)),T)) = ((Red (p2,T)) + (- (Low (p2,T,(m + 1))))) . (HT ((Low (p2,T,m)),T)) by POLYNOM1:def_6
.= ((Red (p2,T)) . (HT ((Low (p2,T,m)),T))) + ((- (Low (p2,T,(m + 1)))) . (HT ((Low (p2,T,m)),T))) by POLYNOM1:15
.= ((Red (p2,T)) . (HT ((Low (p2,T,m)),T))) + (- ((Low (p2,T,(m + 1))) . (HT ((Low (p2,T,m)),T)))) by POLYNOM1:17
.= ((Red (p2,T)) . (HT ((Low (p2,T,m)),T))) + (- (0. L)) by A88, POLYNOM1:def_3
.= ((Red (p2,T)) . (HT ((Low (p2,T,m)),T))) + (0. L) by RLVECT_1:12
.= (Red (p2,T)) . (HT ((Low (p2,T,m)),T)) by RLVECT_1:def_4 ;
A90: HT ((Low (p2,T,m)),T) in Support (Low (p2,T,m)) by A87, XBOOLE_0:def_5;
then A91: HT ((Low (p2,T,m)),T) in Lower_Support (p2,T,m) by A14, Lm3;
A92: Support (Low (p2,T,m)) c= Support p2 by A14, Th26;
now__::_thesis:_not_HT_((Low_(p2,T,m)),T)_=_HT_(p2,T)
assume A93: HT ((Low (p2,T,m)),T) = HT (p2,T) ; ::_thesis: contradiction
A94: now__::_thesis:_for_u_being_set_st_u_in_Support_p2_holds_
u_in_Support_(Low_(p2,T,m))
let u be set ; ::_thesis: ( u in Support p2 implies u in Support (Low (p2,T,m)) )
assume A95: u in Support p2 ; ::_thesis: u in Support (Low (p2,T,m))
then reconsider u9 = u as Element of Bags n ;
 u9 <= HT ((Low (p2,T,m)),T),T by A93, A95, TERMORD:def_6;
then  u9 in Lower_Support (p2,T,m) by A14, A91, A95, Th24;
hence  u in Support (Low (p2,T,m)) by A14, Lm3; ::_thesis: verum
end;
 for u being set st u in Support (Low (p2,T,m)) holds
u in Support p2 by A92;
then  k + 1 = card (Support p2) by A15, A94, TARSKI:1;
hence  contradiction by A9; ::_thesis: verum
end;
then A96: not HT ((Low (p2,T,m)),T) in {(HT (p2,T))} by TARSKI:def_1;
 Support (Red (p2,T)) = (Support p2) \ {(HT (p2,T))} by TERMORD:36;
then  HT ((Low (p2,T,m)),T) in Support red2 by A90, A92, A96, XBOOLE_0:def_5;
then  z1 . (HT ((Low (p2,T,m)),T)) <> 0. L by A89, POLYNOM1:def_3;
then A97: HT ((Low (p2,T,m)),T) in Support z1 by POLYNOM1:def_3;
 Support red2 c= Support p2 by TERMORD:35;
then  (term lowzw) / (HT (p1,T)) in Lower_Support (p2,T,m) by A14, A56, A84, A91, A97, Th24;
then A98: (term lowzw) / (HT (p1,T)) in Support (Low (p2,T,m)) by A14, Lm3;
then  (term lowzw) / (HT (p1,T)) in (Support (Low (p2,T,m))) \ (Support (Low (p2,T,(m + 1)))) by A59, XBOOLE_0:def_5;
then  (term lowzw) / (HT (p1,T)) in {(HT ((Low (p2,T,m)),T))} by A16, Th42;
then A99: (term lowzw) / (HT (p1,T)) = HT ((Low (p2,T,m)),T) by TARSKI:def_1;
then A100: (HM ((Low (p2,T,m)),T)) . (HT ((Low (p2,T,m)),T)) = (Low (p2,T,m)) . ((term lowzw) / (HT (p1,T))) by TERMORD:18
.= p2 . ((term lowzw) / (HT (p1,T))) by A14, A98, Th31 ;
HC ((Low (p2,T,m)),T) = (Low (p2,T,m)) . (HT ((Low (p2,T,m)),T)) by TERMORD:def_7
.= p2 . ((term lowzw) / (HT (p1,T))) by A100, TERMORD:18 ;
hence  Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))) = HM ((Low (p2,T,m)),T) by A99, TERMORD:def_8; ::_thesis: verum
end;
then A101: Low (p2,T,m) = (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) + (Red ((Low (p2,T,m)),T)) by TERMORD:38
.= (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) + (Low (p2,T,(m + 1))) by A16, Th43 ;
A102: ((HM (p1,T)) *' z1) . (term lowzw) <> 0. L by A47, A46, POLYNOM1:def_3;
now__::_thesis:_not_((((HM_(p2,T))_*'_(Red_(p1,T)))_-_((HM_(p1,T))_*'_((Red_(p2,T))_-_(Low_(p2,T,(m_+_1))))))_+_((Red_(p1,T))_*'_(Low_(p2,T,(m_+_1)))))_._(term_lowzw)_=_0._L
assume  ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw) = 0. L ; ::_thesis: contradiction
then  ((HM (p1,T)) *' z1) . (term lowzw) = - (0. L) by A79, RLVECT_1:17;
hence  contradiction by A102, RLVECT_1:12; ::_thesis: verum
end;
then A103: ( p1 <> 0_ (n,L) & term lowzw in Support ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) ) by A61, POLYNOM1:def_3, POLYNOM7:def_1;
((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) = ((((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) + (red1 *' (Low (p2,T,(m + 1))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) by POLYNOM1:def_6
.= ((((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T)))) + (0. L) by A27, POLYNOM1:15
.= (((HM (p2,T)) *' red1) + (- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) by RLVECT_1:def_4
.= (((HM (p2,T)) *' red1) . ((HT ((HM (p2,T)),T)) + (HT (red1,T)))) + ((- ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1)))))) . ((HT ((HM (p2,T)),T)) + (HT (red1,T)))) by POLYNOM1:15
.= (((HM (p2,T)) *' red1) . ((HT ((HM (p2,T)),T)) + (HT (red1,T)))) + (0. L) by A38, POLYNOM1:17
.= ((HM (p2,T)) *' red1) . ((HT ((HM (p2,T)),T)) + (HT (red1,T))) by RLVECT_1:def_4 ;
then  (HT ((HM (p2,T)),T)) + (HT (red1,T)) in Support ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) by A11, A78, POLYNOM1:def_3;
then  (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) <> 0_ (n,L) by POLYNOM7:1;
then  (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) reduces_to ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)),p1, term lowzw,T by A53, A103, POLYRED:def_5;
then A104: (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) reduces_to ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)),p1,T by POLYRED:def_6;
 p1 in {p1} by TARSKI:def_1;
then  (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) reduces_to ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)),{p1},T by A104, POLYRED:def_7;
then  [((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))),(((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)))] in PolyRedRel ({p1},T) by POLYRED:def_13;
then A105: PolyRedRel ({p1},T) reduces (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))),((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)) by REWRITE1:15;
 m + 1 <= card (Support p2) by A16, NAT_1:13;
then A106: PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1)))) by A10, A17;
A107: HT (p1,T) = HT ((HM (p1,T)),T) by TERMORD:26
.= term (HM (p1,T)) by TERMORD:23 ;
 (term lowzw) / (HT (p1,T)) is Element of Bags n by PRE_POLY:def_12;
then A108: (Low (p2,T,(m + 1))) . ((term lowzw) / (HT (p1,T))) = 0. L by A59, POLYNOM1:def_3;
ba1: Low (p2,T,m) = - (- (Low (p2,T,m))) by POLYNOM1:19;
ba2: Low (p2,T,(m + 1)) = - (- (Low (p2,T,(m + 1)))) by POLYNOM1:19;
((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) . (term lowzw) = ((HM (p1,T)) *' (red2 - (Low (p2,T,(m + 1))))) . (((term lowzw) / (HT (p1,T))) + (HT (p1,T))) by A52, GROEB_2:def_1
.= ((HM (p1,T)) . (HT (p1,T))) * ((red2 - (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T)))) by A107, POLYRED:7
.= (p1 . (HT (p1,T))) * ((red2 - (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T)))) by TERMORD:18
.= (HC (p1,T)) * ((red2 - (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T)))) by TERMORD:def_7
.= (HC (p1,T)) * ((red2 + (- (Low (p2,T,(m + 1))))) . ((term lowzw) / (HT (p1,T)))) by POLYNOM1:def_6
.= (HC (p1,T)) * ((red2 . ((term lowzw) / (HT (p1,T)))) + ((- (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T))))) by POLYNOM1:15
.= (HC (p1,T)) * ((p2 . ((term lowzw) / (HT (p1,T)))) + ((- (Low (p2,T,(m + 1)))) . ((term lowzw) / (HT (p1,T))))) by A56, A12, A57, TERMORD:40
.= (HC (p1,T)) * ((p2 . ((term lowzw) / (HT (p1,T)))) + (- ((Low (p2,T,(m + 1))) . ((term lowzw) / (HT (p1,T)))))) by POLYNOM1:17
.= (HC (p1,T)) * ((p2 . ((term lowzw) / (HT (p1,T)))) + (0. L)) by A108, RLVECT_1:12
.= (HC (p1,T)) * (p2 . ((term lowzw) / (HT (p1,T)))) by RLVECT_1:def_4 ;
then ((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1) = (((HC (p1,T)) * (- (p2 . ((term lowzw) / (HT (p1,T)))))) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1) by A79, VECTSP_1:9
.= (((HC (p1,T)) * (- (p2 . ((term lowzw) / (HT (p1,T)))))) * ((HC (p1,T)) ")) * (((term lowzw) / (HT (p1,T))) *' p1) by VECTSP_1:def_11
.= ((- (p2 . ((term lowzw) / (HT (p1,T))))) * ((HC (p1,T)) * ((HC (p1,T)) "))) * (((term lowzw) / (HT (p1,T))) *' p1) by GROUP_1:def_3
.= ((- (p2 . ((term lowzw) / (HT (p1,T))))) * (1. L)) * (((term lowzw) / (HT (p1,T))) *' p1) by VECTSP_1:def_10
.= (- (p2 . ((term lowzw) / (HT (p1,T))))) * (((term lowzw) / (HT (p1,T))) *' p1) by VECTSP_1:def_6 ;
then ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) - (((((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) . (term lowzw)) / (HC (p1,T))) * (((term lowzw) / (HT (p1,T))) *' p1)) = ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + (- ((- (p2 . ((term lowzw) / (HT (p1,T))))) * (((term lowzw) / (HT (p1,T))) *' p1))) by POLYNOM1:def_6
.= ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((- (- (p2 . ((term lowzw) / (HT (p1,T)))))) * (((term lowzw) / (HT (p1,T))) *' p1)) by POLYRED:9
.= ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((p2 . ((term lowzw) / (HT (p1,T)))) * (((term lowzw) / (HT (p1,T))) *' p1)) by RLVECT_1:17
.= ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' p1) by POLYRED:22
.= ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' ((HM (p1,T)) + (Red (p1,T)))) by TERMORD:38
.= ((((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + (((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T)))) by POLYNOM1:26
.= ((((HM (p2,T)) *' (Red (p1,T))) + (- ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1))))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + (((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T)))) by POLYNOM1:def_6
.= (((((HM (p2,T)) *' (Red (p1,T))) + (- ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1))))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T)))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T))) by POLYNOM1:21
.= (((((HM (p2,T)) *' (Red (p1,T))) + (- ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,(m + 1))))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T)))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T))) by POLYNOM1:21
.= (((((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' (- ((Red (p2,T)) - (Low (p2,T,(m + 1))))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T)))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T))) by POLYRED:6
.= ((((HM (p2,T)) *' (Red (p1,T))) + (((HM (p1,T)) *' (- ((Red (p2,T)) - (Low (p2,T,(m + 1)))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (HM (p1,T))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T))) by POLYNOM1:21
.= ((((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' ((- ((Red (p2,T)) - (Low (p2,T,(m + 1))))) + (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))))))) + ((Red (p1,T)) *' (Low (p2,T,(m + 1))))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T))) by POLYNOM1:26
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' ((- ((Red (p2,T)) - (Low (p2,T,(m + 1))))) + (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))))))) + (((Red (p1,T)) *' (Low (p2,T,(m + 1)))) + ((Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T))))) *' (Red (p1,T)))) by POLYNOM1:21
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' ((- ((Red (p2,T)) - (Low (p2,T,(m + 1))))) + (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by A101, POLYNOM1:26
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' ((- ((Red (p2,T)) + (- (Low (p2,T,(m + 1)))))) + (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYNOM1:def_6
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' (((- (Red (p2,T))) + (- (- (Low (p2,T,(m + 1)))))) + (Monom ((p2 . ((term lowzw) / (HT (p1,T)))),((term lowzw) / (HT (p1,T)))))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYRED:1
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' ((- (Red (p2,T))) + (- (- (Low (p2,T,m))))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by ba1, ba2, A101, POLYNOM1:21
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' (- ((Red (p2,T)) + (- (Low (p2,T,m))))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYRED:1
.= (((HM (p2,T)) *' (Red (p1,T))) + ((HM (p1,T)) *' (- ((Red (p2,T)) - (Low (p2,T,m)))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYNOM1:def_6
.= (((HM (p2,T)) *' (Red (p1,T))) + (- ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m)))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYRED:6
.= (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by POLYNOM1:def_6 ;
hence  PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by A105, A106, REWRITE1:16; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
hence  ( not S1[k] or S1[k + 1] ) ; ::_thesis: verum
end;
A109: S1[ 0 ]
proof
let m be Element of NAT ; ::_thesis: ( m <= card (Support p2) & card (Support (Low (p2,T,m))) = 0 implies PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) )
assume that
 m <= card (Support p2) and
A110: card (Support (Low (p2,T,m))) = 0 ; ::_thesis: PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m)))
 Support (Low (p2,T,m)) = {} by A110;
then  Low (p2,T,m) = 0_ (n,L) by POLYNOM7:1;
then (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) = (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))) + ((Red (p1,T)) *' (0_ (n,L))) by POLYRED:4
.= (((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T)))) + (0_ (n,L)) by POLYRED:5
.= ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))) by POLYRED:2 ;
hence  PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),(((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' ((Red (p2,T)) - (Low (p2,T,m))))) + ((Red (p1,T)) *' (Low (p2,T,m))) by REWRITE1:12; ::_thesis: verum
end;
 for i being Element of NAT st 0 <= i & i <= j9 holds
S1[i] from INT_1:sch_7(A109, A8);
hence  PolyRedRel ({p1},T) reduces ((HM (p2,T)) *' (Red (p1,T))) - ((HM (p1,T)) *' (Red (p2,T))),p2 *' (Red (p1,T)) by A3, A7, A4; ::_thesis: verum
end;

theorem Th56: :: GROEB_3:56
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L st HT (p1,T), HT (p2,T) are_disjoint holds
PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
let p1, p2 be Polynomial of n,L; ::_thesis: ( HT (p1,T), HT (p2,T) are_disjoint implies PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) )
assume A1: HT (p1,T), HT (p2,T) are_disjoint ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then A2: S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - ((Red (p2,T)) *' p1) by Th54;
now__::_thesis:_(_(_p1_=_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_or_(_p1_<>_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_)
percases ( p1 = 0_ (n,L) or p1 <> 0_ (n,L) ) ;
case p1 = 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then  ( (Red (p2,T)) *' p1 = 0_ (n,L) & Red (p1,T) = 0_ (n,L) ) by Th11, POLYNOM1:28;
then  S-Poly (p1,p2,T) = (0_ (n,L)) - (0_ (n,L)) by A2, POLYNOM1:28
.= 0_ (n,L) by POLYRED:4 ;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
case p1 <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then reconsider p1a = p1 as non-zero Polynomial of n,L by POLYNOM7:def_1;
now__::_thesis:_for_u_being_set_st_u_in_{p2}_holds_
u_in_{p1,p2}
let u be set ; ::_thesis: ( u in {p2} implies u in {p1,p2} )
assume  u in {p2} ; ::_thesis: u in {p1,p2}
then  u = p2 by TARSKI:def_1;
hence  u in {p1,p2} by TARSKI:def_2; ::_thesis: verum
end;
then A3: {p2} c= {p1,p2} by TARSKI:def_3;
then A4: PolyRedRel ({p2},T) c= PolyRedRel ({p1,p2},T) by GROEB_1:4;
now__::_thesis:_for_u_being_set_st_u_in_{p1}_holds_
u_in_{p1,p2}
let u be set ; ::_thesis: ( u in {p1} implies u in {p1,p2} )
assume  u in {p1} ; ::_thesis: u in {p1,p2}
then  u = p1 by TARSKI:def_1;
hence  u in {p1,p2} by TARSKI:def_2; ::_thesis: verum
end;
then A5: {p1} c= {p1,p2} by TARSKI:def_3;
then A6: PolyRedRel ({p1},T) c= PolyRedRel ({p1,p2},T) by GROEB_1:4;
now__::_thesis:_(_(_p2_=_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_or_(_p2_<>_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_)
percases ( p2 = 0_ (n,L) or p2 <> 0_ (n,L) ) ;
case p2 = 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then  ( (Red (p1,T)) *' p2 = 0_ (n,L) & Red (p2,T) = 0_ (n,L) ) by Th11, POLYNOM1:28;
then  S-Poly (p1,p2,T) = (0_ (n,L)) - (0_ (n,L)) by A2, POLYNOM1:28
.= 0_ (n,L) by POLYRED:4 ;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
case p2 <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then reconsider p2a = p2 as non-zero Polynomial of n,L by POLYNOM7:def_1;
now__::_thesis:_(_(_Red_(p1,T)_=_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_or_(_Red_(p1,T)_<>_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_)
percases ( Red (p1,T) = 0_ (n,L) or Red (p1,T) <> 0_ (n,L) ) ;
case Red (p1,T) = 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then A7: S-Poly (p1,p2,T) = (0_ (n,L)) - ((Red (p2,T)) *' p1) by A2, POLYNOM1:28;
now__::_thesis:_(_(_Red_(p2,T)_=_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_or_(_Red_(p2,T)_<>_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_)
percases ( Red (p2,T) = 0_ (n,L) or Red (p2,T) <> 0_ (n,L) ) ;
case Red (p2,T) = 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then  S-Poly (p1,p2,T) = (0_ (n,L)) - (0_ (n,L)) by A7, POLYNOM1:28
.= 0_ (n,L) by POLYRED:4 ;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
case Red (p2,T) <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then reconsider rp2 = Red (p2,T) as non-zero Polynomial of n,L by POLYNOM7:def_1;
 PolyRedRel ({p1a},T) reduces - (rp2 *' p1a), - (0_ (n,L)) by Th49, Th51;
then  PolyRedRel ({p1a},T) reduces - (rp2 *' p1a), 0_ (n,L) by Th13;
then  PolyRedRel ({p1},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by A7, Th14;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by A5, GROEB_1:4, REWRITE1:22; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) ; ::_thesis: verum
end;
case Red (p1,T) <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then reconsider rp1 = Red (p1,T) as non-zero Polynomial of n,L by POLYNOM7:def_1;
now__::_thesis:_(_(_Red_(p2,T)_=_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_or_(_Red_(p2,T)_<>_0__(n,L)_&_PolyRedRel_({p1,p2},T)_reduces_S-Poly_(p1,p2,T),_0__(n,L)_)_)
percases ( Red (p2,T) = 0_ (n,L) or Red (p2,T) <> 0_ (n,L) ) ;
case Red (p2,T) = 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then  (Red (p2,T)) *' p1 = 0_ (n,L) by POLYNOM1:28;
then A8: S-Poly (p1,p2,T) = ((Red (p1,T)) *' p2) - (0_ (n,L)) by A1, Th54
.= (Red (p1,T)) *' p2 by POLYRED:4 ;
 PolyRedRel ({p2a},T) reduces rp1 *' p2a, 0_ (n,L) by Th51;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by A3, A8, GROEB_1:4, REWRITE1:22; ::_thesis: verum
end;
case Red (p2,T) <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L)
then reconsider rp2 = Red (p2,T) as non-zero Polynomial of n,L by POLYNOM7:def_1;
 S-Poly (p1,p2,T) = ((HM (p2a,T)) *' rp1) - ((HM (p1a,T)) *' rp2) by A1, Th53;
then A9: PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T),p2 *' (Red (p1,T)) by A1, A6, Th55, REWRITE1:22;
 PolyRedRel ({p1,p2},T) reduces rp1 *' p2a, 0_ (n,L) by A4, Th51, REWRITE1:22;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) by A9, REWRITE1:16; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) ; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) ; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) ; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel ({p1,p2},T) reduces S-Poly (p1,p2,T), 0_ (n,L) ; ::_thesis: verum
end;

theorem :: GROEB_3:57
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_wrt T implies for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) )
assume  G is_Groebner_basis_wrt T ; ::_thesis: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
then  for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) by GROEB_2:23;
hence  for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by GROEB_2:24; ::_thesis: verum
end;

theorem :: GROEB_3:58
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non degenerated non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L)
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non degenerated non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non degenerated non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L)
let L be non degenerated non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds
for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L)
let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) implies for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) )
assume A1: not 0_ (n,L) in G ; ::_thesis: ( ex g1, g2 being Polynomial of n,L st
( g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & not PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) or for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) )
assume A2: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; ::_thesis: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L)
 for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
proof
let g1, g2 be Polynomial of n,L; ::_thesis: ( g1 in G & g2 in G implies PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) )
assume that
A3: g1 in G and
A4: g2 in G ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
now__::_thesis:_(_(_HT_(g1,T),_HT_(g2,T)_are_disjoint_&_PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L)_)_or_(_not_HT_(g1,T),_HT_(g2,T)_are_disjoint_&_PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L)_)_)
percases ( HT (g1,T), HT (g2,T) are_disjoint or not HT (g1,T), HT (g2,T) are_disjoint ) ;
caseA5: HT (g1,T), HT (g2,T) are_disjoint ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
now__::_thesis:_for_u_being_set_st_u_in_{g1,g2}_holds_
u_in_G
let u be set ; ::_thesis: ( u in {g1,g2} implies u in G )
assume A6: u in {g1,g2} ; ::_thesis: u in G
now__::_thesis:_(_(_u_=_g1_&_u_in_G_)_or_(_u_=_g2_&_u_in_G_)_)
percases ( u = g1 or u = g2 ) by A6, TARSKI:def_2;
case u = g1 ; ::_thesis: u in G
hence  u in G by A3; ::_thesis: verum
end;
case u = g2 ; ::_thesis: u in G
hence  u in G by A4; ::_thesis: verum
end;
end;
end;
hence  u in G ; ::_thesis: verum
end;
then A7: {g1,g2} c= G by TARSKI:def_3;
 PolyRedRel ({g1,g2},T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A5, Th56;
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A7, GROEB_1:4, REWRITE1:22; ::_thesis: verum
end;
case not HT (g1,T), HT (g2,T) are_disjoint ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A2, A3, A4; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; ::_thesis: verum
end;
then  G is_Groebner_basis_wrt T by A1, GROEB_2:25;
hence  for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) by GROEB_2:23; ::_thesis: verum
end;

theorem :: GROEB_3:59
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
G is_Groebner_basis_wrt T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
G is_Groebner_basis_wrt T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
G is_Groebner_basis_wrt T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) holds
G is_Groebner_basis_wrt T
let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( not 0_ (n,L) in G & ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ) implies G is_Groebner_basis_wrt T )
assume A1: not 0_ (n,L) in G ; ::_thesis: ( ex g1, g2, h being Polynomial of n,L st
( g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) & not h = 0_ (n,L) ) or G is_Groebner_basis_wrt T )
assume A2: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & not HT (g1,T), HT (g2,T) are_disjoint & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds
h = 0_ (n,L) ; ::_thesis: G is_Groebner_basis_wrt T
 for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds
PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
proof
let g1, g2 be Polynomial of n,L; ::_thesis: ( g1 in G & g2 in G implies PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) )
assume that
A3: g1 in G and
A4: g2 in G ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
now__::_thesis:_(_(_HT_(g1,T),_HT_(g2,T)_are_disjoint_&_PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L)_)_or_(_not_HT_(g1,T),_HT_(g2,T)_are_disjoint_&_PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L)_)_)
percases ( HT (g1,T), HT (g2,T) are_disjoint or not HT (g1,T), HT (g2,T) are_disjoint ) ;
caseA5: HT (g1,T), HT (g2,T) are_disjoint ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
now__::_thesis:_for_u_being_set_st_u_in_{g1,g2}_holds_
u_in_G
let u be set ; ::_thesis: ( u in {g1,g2} implies u in G )
assume A6: u in {g1,g2} ; ::_thesis: u in G
now__::_thesis:_(_(_u_=_g1_&_u_in_G_)_or_(_u_=_g2_&_u_in_G_)_)
percases ( u = g1 or u = g2 ) by A6, TARSKI:def_2;
case u = g1 ; ::_thesis: u in G
hence  u in G by A3; ::_thesis: verum
end;
case u = g2 ; ::_thesis: u in G
hence  u in G by A4; ::_thesis: verum
end;
end;
end;
hence  u in G ; ::_thesis: verum
end;
then A7: {g1,g2} c= G by TARSKI:def_3;
 PolyRedRel ({g1,g2},T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A5, Th56;
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A7, GROEB_1:4, REWRITE1:22; ::_thesis: verum
end;
caseA8: not HT (g1,T), HT (g2,T) are_disjoint ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L)
 S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T)
proof
now__::_thesis:_(_(_not_S-Poly_(g1,g2,T)_in_field_(PolyRedRel_(G,T))_&_S-Poly_(g1,g2,T)_has_a_normal_form_wrt_PolyRedRel_(G,T)_)_or_(_S-Poly_(g1,g2,T)_in_field_(PolyRedRel_(G,T))_&_S-Poly_(g1,g2,T)_has_a_normal_form_wrt_PolyRedRel_(G,T)_)_)
percases ( not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) or S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ) ;
case not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; ::_thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T)
hence  S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:46; ::_thesis: verum
end;
case S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; ::_thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T)
hence  S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:def_14; ::_thesis: verum
end;
end;
end;
hence  S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) ; ::_thesis: verum
end;
then consider h being set  such that
A9: h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) by REWRITE1:def_11;
 PolyRedRel (G,T) reduces S-Poly (g1,g2,T),h by A9, REWRITE1:def_6;
then reconsider h = h as Polynomial of n,L by Lm1;
 h = 0_ (n,L) by A2, A3, A4, A8, A9;
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A9, REWRITE1:def_6; ::_thesis: verum
end;
end;
end;
hence  PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; ::_thesis: verum
end;
hence  G is_Groebner_basis_wrt T by A1, GROEB_2:25; ::_thesis: verum
end;