:: ANPROJ_2 semantic presentation

begin


theorem Th1: :: ANPROJ_2:1
for V being RealLinearSpace
for u, v, w being Element of V st ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) holds
( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
proof
let V be RealLinearSpace; ::_thesis: for u, v, w being Element of V st ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) holds
( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
let u, v, w be Element of V; ::_thesis: ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) implies ( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v ) )
assume A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; ::_thesis: ( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
A2: now__::_thesis:_not_v_is_zero
assume  v is zero ; ::_thesis: contradiction
then A3: v = 0. V by STRUCT_0:def_12;
0. V = (0. V) + (0. V) by RLVECT_1:4
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:4
.= ((0. V) + (1 * v)) + (0. V) by A3, RLVECT_1:def_8
.= ((0 * u) + (1 * v)) + (0. V) by RLVECT_1:10
.= ((0 * u) + (1 * v)) + (0 * w) by RLVECT_1:10 ;
hence  contradiction by A1; ::_thesis: verum
end;
A4: now__::_thesis:_not_w_is_zero
assume  w is zero ; ::_thesis: contradiction
then A5: w = 0. V by STRUCT_0:def_12;
0. V = (0. V) + (0. V) by RLVECT_1:4
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:4
.= ((0. V) + (0. V)) + (1 * w) by A5, RLVECT_1:def_8
.= ((0 * u) + (0. V)) + (1 * w) by RLVECT_1:10
.= ((0 * u) + (0 * v)) + (1 * w) by RLVECT_1:10 ;
hence  contradiction by A1; ::_thesis: verum
end;
now__::_thesis:_not_u_is_zero
assume  u is zero ; ::_thesis: contradiction
then A6: u = 0. V by STRUCT_0:def_12;
0. V = (0. V) + (0. V) by RLVECT_1:4
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:4
.= ((1 * u) + (0. V)) + (0. V) by A6, RLVECT_1:def_8
.= ((1 * u) + (0 * v)) + (0. V) by RLVECT_1:10
.= ((1 * u) + (0 * v)) + (0 * w) by RLVECT_1:10 ;
hence  contradiction by A1; ::_thesis: verum
end;
hence  ( not u is zero & not v is zero & not w is zero ) by A2, A4; ::_thesis: ( not u,v,w are_LinDep & not are_Prop u,v )
thus  not u,v,w are_LinDep by A1, ANPROJ_1:def_2; ::_thesis: not are_Prop u,v
hence  not are_Prop u,v by ANPROJ_1:12; ::_thesis: verum
end;

Lm1:  for V being RealLinearSpace
for u, v being Element of V st ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v )
proof
let V be RealLinearSpace; ::_thesis: for u, v being Element of V st ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v )
let u, v be Element of V; ::_thesis: ( ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) implies ( not u is zero & not v is zero & not are_Prop u,v ) )
assume A1: for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ; ::_thesis: ( not u is zero & not v is zero & not are_Prop u,v )
A2: now__::_thesis:_not_v_is_zero
assume  v is zero ; ::_thesis: contradiction
then A3: v = 0. V by STRUCT_0:def_12;
0. V = (0. V) + (0. V) by RLVECT_1:4
.= (0. V) + (1 * v) by A3, RLVECT_1:def_8
.= (0 * u) + (1 * v) by RLVECT_1:10 ;
hence  contradiction by A1; ::_thesis: verum
end;
now__::_thesis:_not_u_is_zero
assume  u is zero ; ::_thesis: contradiction
then A4: u = 0. V by STRUCT_0:def_12;
0. V = (0. V) + (0. V) by RLVECT_1:4
.= (1 * u) + (0. V) by A4, RLVECT_1:def_8
.= (1 * u) + (0 * v) by RLVECT_1:10 ;
hence  contradiction by A1; ::_thesis: verum
end;
hence  ( not u is zero & not v is zero ) by A2; ::_thesis: not are_Prop u,v
given a, b being Real such that A5: a * u = b * v and
A6: a <> 0 and
 b <> 0 ; :: according to ANPROJ_1:def_1 ::_thesis: contradiction
0. V = (a * u) - (b * v) by A5, RLVECT_1:15
.= (a * u) + (b * (- v)) by RLVECT_1:25
.= (a * u) + ((- b) * v) by RLVECT_1:24 ;
hence  contradiction by A1, A6; ::_thesis: verum
end;

theorem Th2: :: ANPROJ_2:2
for V being RealLinearSpace
for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
proof
let V be RealLinearSpace; ::_thesis: for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
let u, v, u1, v1 be Element of V; ::_thesis: ( ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) implies ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep ) )
assume A1: for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
A2: now__::_thesis:_for_d1,_d2,_d3_being_Real_st_((d1_*_u)_+_(d2_*_v))_+_(d3_*_u1)_=_0._V_holds_
(_d1_=_0_&_d2_=_0_&_d3_=_0_)
let d1, d2, d3 be Real; ::_thesis: ( ((d1 * u) + (d2 * v)) + (d3 * u1) = 0. V implies ( d1 = 0 & d2 = 0 & d3 = 0 ) )
assume  ((d1 * u) + (d2 * v)) + (d3 * u1) = 0. V ; ::_thesis: ( d1 = 0 & d2 = 0 & d3 = 0 )
then  0. V = (((d1 * u) + (d2 * v)) + (d3 * u1)) + (0. V) by RLVECT_1:4
.= (((d1 * u) + (d2 * v)) + (d3 * u1)) + (0 * v1) by RLVECT_1:10 ;
hence  ( d1 = 0 & d2 = 0 & d3 = 0 ) by A1; ::_thesis: verum
end;
now__::_thesis:_for_d1,_d2,_d3_being_Real_st_((d1_*_u1)_+_(d2_*_v1))_+_(d3_*_u)_=_0._V_holds_
(_d1_=_0_&_d2_=_0_&_d3_=_0_)
let d1, d2, d3 be Real; ::_thesis: ( ((d1 * u1) + (d2 * v1)) + (d3 * u) = 0. V implies ( d1 = 0 & d2 = 0 & d3 = 0 ) )
assume  ((d1 * u1) + (d2 * v1)) + (d3 * u) = 0. V ; ::_thesis: ( d1 = 0 & d2 = 0 & d3 = 0 )
then  0. V = ((d3 * u) + (d1 * u1)) + (d2 * v1) by RLVECT_1:def_3
.= (((d3 * u) + (0. V)) + (d1 * u1)) + (d2 * v1) by RLVECT_1:4
.= (((d3 * u) + (0 * v)) + (d1 * u1)) + (d2 * v1) by RLVECT_1:10 ;
hence  ( d1 = 0 & d2 = 0 & d3 = 0 ) by A1; ::_thesis: verum
end;
hence  ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep ) by A2, Th1; ::_thesis: verum
end;

Lm2:  for V being RealLinearSpace
for v, w, u being Element of V
for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
proof
let V be RealLinearSpace; ::_thesis: for v, w, u being Element of V
for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
let v, w, u be Element of V; ::_thesis: for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
let a, b, c, d be Real; ::_thesis: a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
thus (((a * b) * v) + ((a * c) * w)) + ((a * d) * u) = ((a * (b * v)) + ((a * c) * w)) + ((a * d) * u) by RLVECT_1:def_7
.= ((a * (b * v)) + (a * (c * w))) + ((a * d) * u) by RLVECT_1:def_7
.= (a * ((b * v) + (c * w))) + ((a * d) * u) by RLVECT_1:def_5
.= (a * ((b * v) + (c * w))) + (a * (d * u)) by RLVECT_1:def_7
.= a * (((b * v) + (c * w)) + (d * u)) by RLVECT_1:def_5 ; ::_thesis: verum
end;

Lm3:  for V being RealLinearSpace
for u, v, w, u1, v1, w1 being Element of V holds ((u + v) + w) + ((u1 + v1) + w1) = ((u + u1) + (v + v1)) + (w + w1)
proof
let V be RealLinearSpace; ::_thesis: for u, v, w, u1, v1, w1 being Element of V holds ((u + v) + w) + ((u1 + v1) + w1) = ((u + u1) + (v + v1)) + (w + w1)
let u, v, w, u1, v1, w1 be Element of V; ::_thesis: ((u + v) + w) + ((u1 + v1) + w1) = ((u + u1) + (v + v1)) + (w + w1)
thus ((u + u1) + (v + v1)) + (w + w1) = (u1 + (u + (v + v1))) + (w + w1) by RLVECT_1:def_3
.= (u1 + ((u + v) + v1)) + (w + w1) by RLVECT_1:def_3
.= ((u1 + v1) + (u + v)) + (w + w1) by RLVECT_1:def_3
.= (u1 + v1) + ((u + v) + (w + w1)) by RLVECT_1:def_3
.= (u1 + v1) + (((u + v) + w) + w1) by RLVECT_1:def_3
.= ((u + v) + w) + ((u1 + v1) + w1) by RLVECT_1:def_3 ; ::_thesis: verum
end;

theorem Th3: :: ANPROJ_2:3
for V being RealLinearSpace
for p, q, r being Element of V st ( for w being Element of V ex a, b, c being Real st w = ((a * p) + (b * q)) + (c * r) ) & ( for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) holds
for u, u1 being Element of V ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
proof
let V be RealLinearSpace; ::_thesis: for p, q, r being Element of V st ( for w being Element of V ex a, b, c being Real st w = ((a * p) + (b * q)) + (c * r) ) & ( for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) holds
for u, u1 being Element of V ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
let p, q, r be Element of V; ::_thesis: ( ( for w being Element of V ex a, b, c being Real st w = ((a * p) + (b * q)) + (c * r) ) & ( for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) implies for u, u1 being Element of V ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) )
assume that
A1: for w being Element of V ex a, b, c being Real st w = ((a * p) + (b * q)) + (c * r) and
A2: for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; ::_thesis: for u, u1 being Element of V ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
let u, u1 be Element of V; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
consider a, b, c being Real such that
A3: u = ((a * p) + (b * q)) + (c * r) by A1;
consider a1, b1, c1 being Real such that
A4: u1 = ((a1 * p) + (b1 * q)) + (c1 * r) by A1;
A5: for a3, b3 being Real holds (a3 * u) + (b3 * u1) = ((((a3 * a) + (b3 * a1)) * p) + (((a3 * b) + (b3 * b1)) * q)) + (((a3 * c) + (b3 * c1)) * r)
proof
let a3, b3 be Real; ::_thesis: (a3 * u) + (b3 * u1) = ((((a3 * a) + (b3 * a1)) * p) + (((a3 * b) + (b3 * b1)) * q)) + (((a3 * c) + (b3 * c1)) * r)
 a3 * u = (((a3 * a) * p) + ((a3 * b) * q)) + ((a3 * c) * r) by A3, Lm2;
hence (a3 * u) + (b3 * u1) = ((((a3 * a) * p) + ((a3 * b) * q)) + ((a3 * c) * r)) + ((((b3 * a1) * p) + ((b3 * b1) * q)) + ((b3 * c1) * r)) by A4, Lm2
.= ((((a3 * a) * p) + ((b3 * a1) * p)) + (((a3 * b) * q) + ((b3 * b1) * q))) + (((a3 * c) * r) + ((b3 * c1) * r)) by Lm3
.= ((((a3 * a) + (b3 * a1)) * p) + (((a3 * b) * q) + ((b3 * b1) * q))) + (((a3 * c) * r) + ((b3 * c1) * r)) by RLVECT_1:def_6
.= ((((a3 * a) + (b3 * a1)) * p) + (((a3 * b) + (b3 * b1)) * q)) + (((a3 * c) * r) + ((b3 * c1) * r)) by RLVECT_1:def_6
.= ((((a3 * a) + (b3 * a1)) * p) + (((a3 * b) + (b3 * b1)) * q)) + (((a3 * c) + (b3 * c1)) * r) by RLVECT_1:def_6 ;
::_thesis: verum
end;
A6: not q is zero by A2, Th1;
A7: now__::_thesis:_(_(_are_Prop_u,u1_or_u_is_zero_or_u1_is_zero_)_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
A8: now__::_thesis:_(_u1_is_zero_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
assume  u1 is zero ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
then  u1 = 0. V by STRUCT_0:def_12;
then  ( p,q,q are_LinDep & u,u1,q are_LinDep ) by ANPROJ_1:10, ANPROJ_1:11;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A6; ::_thesis: verum
end;
A9: now__::_thesis:_(_u_is_zero_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
assume  u is zero ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
then  u = 0. V by STRUCT_0:def_12;
then  ( p,q,q are_LinDep & u,u1,q are_LinDep ) by ANPROJ_1:10, ANPROJ_1:11;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A6; ::_thesis: verum
end;
A10: now__::_thesis:_(_are_Prop_u,u1_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
assume  are_Prop u,u1 ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
then  ( p,q,q are_LinDep & u,u1,q are_LinDep ) by ANPROJ_1:11;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A6; ::_thesis: verum
end;
assume  ( are_Prop u,u1 or u is zero or u1 is zero ) ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A10, A9, A8; ::_thesis: verum
end;
A11: ( not p is zero & not are_Prop p,q ) by A2, Th1;
A12: now__::_thesis:_(_not_are_Prop_u,u1_&_not_u_is_zero_&_not_u1_is_zero_&_c_<>_0_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
assume that
A13: not are_Prop u,u1 and
A14: not u is zero and
A15: not u1 is zero and
A16: c <> 0 ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
A17: now__::_thesis:_(_c1_<>_0_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
set a3 = 1;
set b3 = - (c * (c1 "));
set y = (1 * u) + ((- (c * (c1 "))) * u1);
assume A18: c1 <> 0 ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
then  c1 " <> 0 by XCMPLX_1:202;
then A19: c * (c1 ") <> 0 by A16, XCMPLX_1:6;
A20: not (1 * u) + ((- (c * (c1 "))) * u1) is zero
proof
assume  (1 * u) + ((- (c * (c1 "))) * u1) is zero ; ::_thesis: contradiction
then  0. V = (1 * u) + ((- (c * (c1 "))) * u1) by STRUCT_0:def_12
.= (1 * u) + ((c * (c1 ")) * (- u1)) by RLVECT_1:24
.= (1 * u) + (- ((c * (c1 ")) * u1)) by RLVECT_1:25 ;
then  - (1 * u) = - ((c * (c1 ")) * u1) by RLVECT_1:def_10;
then  1 * u = (c * (c1 ")) * u1 by RLVECT_1:18;
hence  contradiction by A13, A19, ANPROJ_1:def_1; ::_thesis: verum
end;
(1 * c) + ((- (c * (c1 "))) * c1) = c + ((- c) * ((c1 ") * c1))
.= c + ((- c) * 1) by A18, XCMPLX_0:def_7
.= 0 ;
then (1 * u) + ((- (c * (c1 "))) * u1) = ((((1 * a) + ((- (c * (c1 "))) * a1)) * p) + (((1 * b) + ((- (c * (c1 "))) * b1)) * q)) + (0 * r) by A5
.= ((((1 * a) + ((- (c * (c1 "))) * a1)) * p) + (((1 * b) + ((- (c * (c1 "))) * b1)) * q)) + (0. V) by RLVECT_1:10
.= (((1 * a) + ((- (c * (c1 "))) * a1)) * p) + (((1 * b) + ((- (c * (c1 "))) * b1)) * q) by RLVECT_1:4 ;
then A21: p,q,(1 * u) + ((- (c * (c1 "))) * u1) are_LinDep by A6, A11, ANPROJ_1:6;
 u,u1,(1 * u) + ((- (c * (c1 "))) * u1) are_LinDep by A13, A14, A15, ANPROJ_1:6;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A20, A21; ::_thesis: verum
end;
now__::_thesis:_(_c1_=_0_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
set a3 = 0 ;
set b3 = 1;
set y = (0 * u) + (1 * u1);
A22: (0 * u) + (1 * u1) = (0 * u) + u1 by RLVECT_1:def_8
.= (0. V) + u1 by RLVECT_1:10
.= u1 by RLVECT_1:4 ;
assume  c1 = 0 ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
then  (0 * c) + (1 * c1) = 0 ;
then (0 * u) + (1 * u1) = ((((0 * a) + (1 * a1)) * p) + (((0 * b) + (1 * b1)) * q)) + (0 * r) by A5
.= ((((0 * a) + (1 * a1)) * p) + (((0 * b) + (1 * b1)) * q)) + (0. V) by RLVECT_1:10
.= (((0 * a) + (1 * a1)) * p) + (((0 * b) + (1 * b1)) * q) by RLVECT_1:4 ;
then A23: p,q,(0 * u) + (1 * u1) are_LinDep by A6, A11, ANPROJ_1:6;
 u,u1,(0 * u) + (1 * u1) are_LinDep by A13, A14, A15, ANPROJ_1:6;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A15, A22, A23; ::_thesis: verum
end;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A17; ::_thesis: verum
end;
now__::_thesis:_(_not_are_Prop_u,u1_&_not_u_is_zero_&_not_u1_is_zero_&_c_=_0_implies_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)_)
assume that
A24: not are_Prop u,u1 and
A25: not u is zero and
A26: not u1 is zero and
A27: c = 0 ; ::_thesis: ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
now__::_thesis:_ex_y_being_Element_of_V_st_
(_p,q,y_are_LinDep_&_u,u1,y_are_LinDep_&_not_y_is_zero_)
set a3 = 1;
set b3 = 0 ;
set y = (1 * u) + (0 * u1);
A28: (1 * u) + (0 * u1) = u + (0 * u1) by RLVECT_1:def_8
.= u + (0. V) by RLVECT_1:10
.= u by RLVECT_1:4 ;
 (1 * c) + (0 * c1) = 0 by A27;
then (1 * u) + (0 * u1) = ((((1 * a) + (0 * a1)) * p) + (((1 * b) + (0 * b1)) * q)) + (0 * r) by A5
.= ((((1 * a) + (0 * a1)) * p) + (((1 * b) + (0 * b1)) * q)) + (0. V) by RLVECT_1:10
.= (((1 * a) + (0 * a1)) * p) + (((1 * b) + (0 * b1)) * q) by RLVECT_1:4 ;
then A29: p,q,(1 * u) + (0 * u1) are_LinDep by A6, A11, ANPROJ_1:6;
 u,u1,(1 * u) + (0 * u1) are_LinDep by A24, A25, A26, ANPROJ_1:6;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A25, A28, A29; ::_thesis: verum
end;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) ; ::_thesis: verum
end;
hence  ex y being Element of V st
( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero ) by A7, A12; ::_thesis: verum
end;

Lm4:  for V being RealLinearSpace
for v, w, u, y being Element of V
for a, b, c, d, d1 being Real holds a * ((((b * v) + (c * w)) + (d * u)) + (d1 * y)) = ((((a * b) * v) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y)
proof
let V be RealLinearSpace; ::_thesis: for v, w, u, y being Element of V
for a, b, c, d, d1 being Real holds a * ((((b * v) + (c * w)) + (d * u)) + (d1 * y)) = ((((a * b) * v) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y)
let v, w, u, y be Element of V; ::_thesis: for a, b, c, d, d1 being Real holds a * ((((b * v) + (c * w)) + (d * u)) + (d1 * y)) = ((((a * b) * v) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y)
let a, b, c, d, d1 be Real; ::_thesis: a * ((((b * v) + (c * w)) + (d * u)) + (d1 * y)) = ((((a * b) * v) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y)
thus ((((a * b) * v) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y) = (((a * (b * v)) + ((a * c) * w)) + ((a * d) * u)) + ((a * d1) * y) by RLVECT_1:def_7
.= (((a * (b * v)) + (a * (c * w))) + ((a * d) * u)) + ((a * d1) * y) by RLVECT_1:def_7
.= ((a * ((b * v) + (c * w))) + ((a * d) * u)) + ((a * d1) * y) by RLVECT_1:def_5
.= ((a * ((b * v) + (c * w))) + (a * (d * u))) + ((a * d1) * y) by RLVECT_1:def_7
.= ((a * ((b * v) + (c * w))) + (a * (d * u))) + (a * (d1 * y)) by RLVECT_1:def_7
.= (a * (((b * v) + (c * w)) + (d * u))) + (a * (d1 * y)) by RLVECT_1:def_5
.= a * ((((b * v) + (c * w)) + (d * u)) + (d1 * y)) by RLVECT_1:def_5 ; ::_thesis: verum
end;

Lm5:  for V being RealLinearSpace
for u, v, w, y, u1, v1, w1, y1 being Element of V holds (((u + v) + w) + y) + (((u1 + v1) + w1) + y1) = (((u + u1) + (v + v1)) + (w + w1)) + (y + y1)
proof
let V be RealLinearSpace; ::_thesis: for u, v, w, y, u1, v1, w1, y1 being Element of V holds (((u + v) + w) + y) + (((u1 + v1) + w1) + y1) = (((u + u1) + (v + v1)) + (w + w1)) + (y + y1)
let u, v, w, y, u1, v1, w1, y1 be Element of V; ::_thesis: (((u + v) + w) + y) + (((u1 + v1) + w1) + y1) = (((u + u1) + (v + v1)) + (w + w1)) + (y + y1)
thus (((u + u1) + (v + v1)) + (w + w1)) + (y + y1) = ((u1 + (u + (v + v1))) + (w + w1)) + (y + y1) by RLVECT_1:def_3
.= ((u1 + ((u + v) + v1)) + (w + w1)) + (y + y1) by RLVECT_1:def_3
.= (((u1 + v1) + (u + v)) + (w + w1)) + (y + y1) by RLVECT_1:def_3
.= ((u1 + v1) + ((u + v) + (w + w1))) + (y + y1) by RLVECT_1:def_3
.= ((u1 + v1) + (((u + v) + w) + w1)) + (y + y1) by RLVECT_1:def_3
.= (((u1 + v1) + w1) + ((u + v) + w)) + (y + y1) by RLVECT_1:def_3
.= ((u + v) + w) + (((u1 + v1) + w1) + (y + y1)) by RLVECT_1:def_3
.= ((u + v) + w) + (y + (y1 + ((u1 + v1) + w1))) by RLVECT_1:def_3
.= (((u + v) + w) + y) + (((u1 + v1) + w1) + y1) by RLVECT_1:def_3 ; ::_thesis: verum
end;

Lm6:  for V being RealLinearSpace
for v, w, u being Element of V
for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
proof
let V be RealLinearSpace; ::_thesis: for v, w, u being Element of V
for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
let v, w, u be Element of V; ::_thesis: for a, b, c, d being Real holds a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
let a, b, c, d be Real; ::_thesis: a * (((b * v) + (c * w)) + (d * u)) = (((a * b) * v) + ((a * c) * w)) + ((a * d) * u)
thus (((a * b) * v) + ((a * c) * w)) + ((a * d) * u) = ((a * (b * v)) + ((a * c) * w)) + ((a * d) * u) by RLVECT_1:def_7
.= ((a * (b * v)) + (a * (c * w))) + ((a * d) * u) by RLVECT_1:def_7
.= (a * ((b * v) + (c * w))) + ((a * d) * u) by RLVECT_1:def_5
.= (a * ((b * v) + (c * w))) + (a * (d * u)) by RLVECT_1:def_7
.= a * (((b * v) + (c * w)) + (d * u)) by RLVECT_1:def_5 ; ::_thesis: verum
end;

Lm7:  for V being RealLinearSpace
for y, p, w, q, r being Element of V
for a1, b1, a, b, c being Real st y = (a1 * p) + (b1 * w) & w = ((a * p) + (b * q)) + (c * r) holds
y = (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r)
proof
let V be RealLinearSpace; ::_thesis: for y, p, w, q, r being Element of V
for a1, b1, a, b, c being Real st y = (a1 * p) + (b1 * w) & w = ((a * p) + (b * q)) + (c * r) holds
y = (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r)
let y, p, w, q, r be Element of V; ::_thesis: for a1, b1, a, b, c being Real st y = (a1 * p) + (b1 * w) & w = ((a * p) + (b * q)) + (c * r) holds
y = (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r)
let a1, b1, a, b, c be Real; ::_thesis: ( y = (a1 * p) + (b1 * w) & w = ((a * p) + (b * q)) + (c * r) implies y = (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r) )
assume  ( y = (a1 * p) + (b1 * w) & w = ((a * p) + (b * q)) + (c * r) ) ; ::_thesis: y = (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r)
hence y = (a1 * p) + ((((b1 * a) * p) + ((b1 * b) * q)) + ((b1 * c) * r)) by Lm6
.= (a1 * p) + (((b1 * a) * p) + (((b1 * b) * q) + ((b1 * c) * r))) by RLVECT_1:def_3
.= ((a1 * p) + ((b1 * a) * p)) + (((b1 * b) * q) + ((b1 * c) * r)) by RLVECT_1:def_3
.= ((a1 + (b1 * a)) * p) + (((b1 * b) * q) + ((b1 * c) * r)) by RLVECT_1:def_6
.= (((a1 + (b1 * a)) * p) + ((b1 * b) * q)) + ((b1 * c) * r) by RLVECT_1:def_3 ;
::_thesis: verum
end;

theorem Th4: :: ANPROJ_2:4
for V being RealLinearSpace
for p, q, r, s being Element of V st ( for w being Element of V ex a, b, c, d being Real st w = (((a * p) + (b * q)) + (c * r)) + (d * s) ) & ( for a, b, c, d being Real st (((a * p) + (b * q)) + (c * r)) + (d * s) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) holds
for u, v being Element of V st not u is zero & not v is zero holds
ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
proof
let V be RealLinearSpace; ::_thesis: for p, q, r, s being Element of V st ( for w being Element of V ex a, b, c, d being Real st w = (((a * p) + (b * q)) + (c * r)) + (d * s) ) & ( for a, b, c, d being Real st (((a * p) + (b * q)) + (c * r)) + (d * s) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) holds
for u, v being Element of V st not u is zero & not v is zero holds
ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
let p, q, r, s be Element of V; ::_thesis: ( ( for w being Element of V ex a, b, c, d being Real st w = (((a * p) + (b * q)) + (c * r)) + (d * s) ) & ( for a, b, c, d being Real st (((a * p) + (b * q)) + (c * r)) + (d * s) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) implies for u, v being Element of V st not u is zero & not v is zero holds
ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) )
assume that
A1: for w being Element of V ex a, b, c, d being Real st w = (((a * p) + (b * q)) + (c * r)) + (d * s) and
A2: for a, b, c, d being Real st (((a * p) + (b * q)) + (c * r)) + (d * s) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ; ::_thesis: for u, v being Element of V st not u is zero & not v is zero holds
ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
A3: not p is zero by A2, Th2;
let u, v be Element of V; ::_thesis: ( not u is zero & not v is zero implies ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) )
assume that
A4: not u is zero and
A5: not v is zero ; ::_thesis: ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
consider a1, b1, c1, d1 being Real such that
A6: u = (((a1 * p) + (b1 * q)) + (c1 * r)) + (d1 * s) by A1;
 not p,q,r are_LinDep by A2, Th2;
then A7: not are_Prop q,r by ANPROJ_1:11;
A8: not q is zero by A2, Th2;
consider a2, b2, c2, d2 being Real such that
A9: v = (((a2 * p) + (b2 * q)) + (c2 * r)) + (d2 * s) by A1;
A10: for a3, b3 being Real holds (a3 * u) + (b3 * v) = (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) + (b3 * b2)) * q)) + (((a3 * c1) + (b3 * c2)) * r)) + (((a3 * d1) + (b3 * d2)) * s)
proof
let a3, b3 be Real; ::_thesis: (a3 * u) + (b3 * v) = (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) + (b3 * b2)) * q)) + (((a3 * c1) + (b3 * c2)) * r)) + (((a3 * d1) + (b3 * d2)) * s)
 a3 * u = ((((a3 * a1) * p) + ((a3 * b1) * q)) + ((a3 * c1) * r)) + ((a3 * d1) * s) by A6, Lm4;
hence (a3 * u) + (b3 * v) = (((((a3 * a1) * p) + ((a3 * b1) * q)) + ((a3 * c1) * r)) + ((a3 * d1) * s)) + (((((b3 * a2) * p) + ((b3 * b2) * q)) + ((b3 * c2) * r)) + ((b3 * d2) * s)) by A9, Lm4
.= (((((a3 * a1) * p) + ((b3 * a2) * p)) + (((a3 * b1) * q) + ((b3 * b2) * q))) + (((a3 * c1) * r) + ((b3 * c2) * r))) + (((a3 * d1) * s) + ((b3 * d2) * s)) by Lm5
.= (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) * q) + ((b3 * b2) * q))) + (((a3 * c1) * r) + ((b3 * c2) * r))) + (((a3 * d1) * s) + ((b3 * d2) * s)) by RLVECT_1:def_6
.= (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) + (b3 * b2)) * q)) + (((a3 * c1) * r) + ((b3 * c2) * r))) + (((a3 * d1) * s) + ((b3 * d2) * s)) by RLVECT_1:def_6
.= (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) + (b3 * b2)) * q)) + (((a3 * c1) + (b3 * c2)) * r)) + (((a3 * d1) * s) + ((b3 * d2) * s)) by RLVECT_1:def_6
.= (((((a3 * a1) + (b3 * a2)) * p) + (((a3 * b1) + (b3 * b2)) * q)) + (((a3 * c1) + (b3 * c2)) * r)) + (((a3 * d1) + (b3 * d2)) * s) by RLVECT_1:def_6 ;
::_thesis: verum
end;
A11: not r is zero by A2, Th2;
A12: now__::_thesis:_(_not_are_Prop_u,v_implies_ex_y,_w_being_Element_of_V_st_
(_u,v,w_are_LinDep_&_q,r,y_are_LinDep_&_p,w,y_are_LinDep_&_not_y_is_zero_&_not_w_is_zero_)_)
assume A13: not are_Prop u,v ; ::_thesis: ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
 ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
proof
A14: now__::_thesis:_(_d1_<>_0_&_d2_<>_0_implies_ex_w_being_Element_of_V_st_
(_not_w_is_zero_&_u,v,w_are_LinDep_&_ex_A,_B,_C_being_Real_st_w_=_((A_*_p)_+_(B_*_q))_+_(C_*_r)_)_)
set a3 = - (d2 * (d1 "));
set b3 = 1;
set w = ((- (d2 * (d1 "))) * u) + (1 * v);
assume that
A15: d1 <> 0 and
A16: d2 <> 0 ; ::_thesis: ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
set A = ((- (d2 * (d1 "))) * a1) + (1 * a2);
set B = ((- (d2 * (d1 "))) * b1) + (1 * b2);
set C = ((- (d2 * (d1 "))) * c1) + (1 * c2);
A17: ( ((- (d2 * (d1 "))) * a1) + (1 * a2) <> 0 or ((- (d2 * (d1 "))) * b1) + (1 * b2) <> 0 or ((- (d2 * (d1 "))) * c1) + (1 * c2) <> 0 )
proof
A18: d2 * (d1 ") <> 0
proof
assume  not d2 * (d1 ") <> 0 ; ::_thesis: contradiction
then  d1 " = 0 by A16, XCMPLX_1:6;
hence  contradiction by A15, XCMPLX_1:202; ::_thesis: verum
end;
A19: (d2 * (d1 ")) * d1 = d2 * ((d1 ") * d1)
.= d2 * 1 by A15, XCMPLX_0:def_7
.= d2 ;
assume A20: ( not ((- (d2 * (d1 "))) * a1) + (1 * a2) <> 0 & not ((- (d2 * (d1 "))) * b1) + (1 * b2) <> 0 & not ((- (d2 * (d1 "))) * c1) + (1 * c2) <> 0 ) ; ::_thesis: contradiction
then A21: - (- ((d2 * (d1 ")) * c1)) = c2 ;
 ( - (- ((d2 * (d1 ")) * a1)) = a2 & - (- ((d2 * (d1 ")) * b1)) = b2 ) by A20;
then  (d2 * (d1 ")) * u = v by A6, A9, A21, A19, Lm4;
hence  contradiction by A13, A18, ANPROJ_1:1; ::_thesis: verum
end;
((- (d2 * (d1 "))) * d1) + (1 * d2) = (- (d2 * ((d1 ") * d1))) + d2
.= (- (d2 * 1)) + d2 by A15, XCMPLX_0:def_7
.= 0 ;
then A22: ((- (d2 * (d1 "))) * u) + (1 * v) = ((((((- (d2 * (d1 "))) * a1) + (1 * a2)) * p) + ((((- (d2 * (d1 "))) * b1) + (1 * b2)) * q)) + ((((- (d2 * (d1 "))) * c1) + (1 * c2)) * r)) + (0 * s) by A10
.= ((((((- (d2 * (d1 "))) * a1) + (1 * a2)) * p) + ((((- (d2 * (d1 "))) * b1) + (1 * b2)) * q)) + ((((- (d2 * (d1 "))) * c1) + (1 * c2)) * r)) + (0. V) by RLVECT_1:10
.= (((((- (d2 * (d1 "))) * a1) + (1 * a2)) * p) + ((((- (d2 * (d1 "))) * b1) + (1 * b2)) * q)) + ((((- (d2 * (d1 "))) * c1) + (1 * c2)) * r) by RLVECT_1:4 ;
then A23: ((- (d2 * (d1 "))) * u) + (1 * v) = ((((((- (d2 * (d1 "))) * a1) + (1 * a2)) * p) + ((((- (d2 * (d1 "))) * b1) + (1 * b2)) * q)) + ((((- (d2 * (d1 "))) * c1) + (1 * c2)) * r)) + (0. V) by RLVECT_1:4
.= ((((((- (d2 * (d1 "))) * a1) + (1 * a2)) * p) + ((((- (d2 * (d1 "))) * b1) + (1 * b2)) * q)) + ((((- (d2 * (d1 "))) * c1) + (1 * c2)) * r)) + (0 * s) by RLVECT_1:10 ;
A24: not ((- (d2 * (d1 "))) * u) + (1 * v) is zero
proof
assume  ((- (d2 * (d1 "))) * u) + (1 * v) is zero ; ::_thesis: contradiction
then  ((- (d2 * (d1 "))) * u) + (1 * v) = 0. V by STRUCT_0:def_12;
hence  contradiction by A2, A23, A17; ::_thesis: verum
end;
 u,v,((- (d2 * (d1 "))) * u) + (1 * v) are_LinDep by A4, A5, A13, ANPROJ_1:6;
hence  ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) ) by A22, A24; ::_thesis: verum
end;
A25: now__::_thesis:_(_d2_=_0_implies_ex_w,_w_being_Element_of_V_st_
(_not_w_is_zero_&_u,v,w_are_LinDep_&_ex_A,_B,_C_being_Real_st_w_=_((A_*_p)_+_(B_*_q))_+_(C_*_r)_)_)
assume A26: d2 = 0 ; ::_thesis: ex w, w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
take w = v; ::_thesis: ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
A27: u,v,w are_LinDep by ANPROJ_1:11;
w = (((a2 * p) + (b2 * q)) + (c2 * r)) + (0. V) by A9, A26, RLVECT_1:10
.= ((a2 * p) + (b2 * q)) + (c2 * r) by RLVECT_1:4 ;
hence  ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) ) by A5, A27; ::_thesis: verum
end;
now__::_thesis:_(_d1_=_0_implies_ex_w,_w_being_Element_of_V_st_
(_not_w_is_zero_&_u,v,w_are_LinDep_&_ex_A,_B,_C_being_Real_st_w_=_((A_*_p)_+_(B_*_q))_+_(C_*_r)_)_)
assume A28: d1 = 0 ; ::_thesis: ex w, w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
take w = u; ::_thesis: ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) )
A29: u,v,w are_LinDep by ANPROJ_1:11;
w = (((a1 * p) + (b1 * q)) + (c1 * r)) + (0. V) by A6, A28, RLVECT_1:10
.= ((a1 * p) + (b1 * q)) + (c1 * r) by RLVECT_1:4 ;
hence  ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) ) by A4, A29; ::_thesis: verum
end;
hence  ex w being Element of V st
( not w is zero & u,v,w are_LinDep & ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) ) by A25, A14; ::_thesis: verum
end;
then consider w being Element of V such that
A30: not w is zero and
A31: u,v,w are_LinDep and
A32: ex A, B, C being Real st w = ((A * p) + (B * q)) + (C * r) ;
consider A, B, C being Real such that
A33: w = ((A * p) + (B * q)) + (C * r) by A32;
A34: now__::_thesis:_(_not_are_Prop_p,w_implies_ex_y,_w_being_Element_of_V_st_
(_u,v,w_are_LinDep_&_q,r,y_are_LinDep_&_p,w,y_are_LinDep_&_not_y_is_zero_&_not_w_is_zero_)_)
set b = 1;
set a = - A;
set y = ((- A) * p) + (1 * w);
A35: ((- A) * p) + (1 * w) = ((((- A) + (1 * A)) * p) + ((1 * B) * q)) + ((1 * C) * r) by A33, Lm7
.= ((0. V) + ((1 * B) * q)) + ((1 * C) * r) by RLVECT_1:10
.= (B * q) + (C * r) by RLVECT_1:4 ;
assume A36: not are_Prop p,w ; ::_thesis: ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
then A37: p,w,((- A) * p) + (1 * w) are_LinDep by A3, A30, ANPROJ_1:6;
A38: ( B <> 0 or C <> 0 )
proof
assume  ( not B <> 0 & not C <> 0 ) ; ::_thesis: contradiction
then A39: w = ((A * p) + (0. V)) + (0 * r) by A33, RLVECT_1:10
.= ((A * p) + (0. V)) + (0. V) by RLVECT_1:10
.= (A * p) + (0. V) by RLVECT_1:4
.= A * p by RLVECT_1:4 ;
 A <> 0
proof
assume  not A <> 0 ; ::_thesis: contradiction
then  w = 0. V by A39, RLVECT_1:10;
hence  contradiction by A30; ::_thesis: verum
end;
hence  contradiction by A36, A39, ANPROJ_1:1; ::_thesis: verum
end;
A40: not ((- A) * p) + (1 * w) is zero
proof
assume  ((- A) * p) + (1 * w) is zero ; ::_thesis: contradiction
then  0. V = (B * q) + (C * r) by A35, STRUCT_0:def_12
.= ((0. V) + (B * q)) + (C * r) by RLVECT_1:4
.= ((0 * p) + (B * q)) + (C * r) by RLVECT_1:10
.= (((0 * p) + (B * q)) + (C * r)) + (0. V) by RLVECT_1:4
.= (((0 * p) + (B * q)) + (C * r)) + (0 * s) by RLVECT_1:10 ;
hence  contradiction by A2, A38; ::_thesis: verum
end;
 q,r,((- A) * p) + (1 * w) are_LinDep by A8, A11, A7, A35, ANPROJ_1:6;
hence  ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) by A30, A31, A40, A37; ::_thesis: verum
end;
now__::_thesis:_(_are_Prop_p,w_implies_ex_y,_w_being_Element_of_V_st_
(_u,v,w_are_LinDep_&_q,r,y_are_LinDep_&_p,w,y_are_LinDep_&_not_y_is_zero_&_not_w_is_zero_)_)
assume  are_Prop p,w ; ::_thesis: ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
then  ( q,r,q are_LinDep & p,w,q are_LinDep ) by ANPROJ_1:11;
hence  ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) by A8, A30, A31; ::_thesis: verum
end;
hence  ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) by A34; ::_thesis: verum
end;
now__::_thesis:_(_are_Prop_u,v_implies_ex_y,_w_being_Element_of_V_st_
(_u,v,w_are_LinDep_&_q,r,y_are_LinDep_&_p,w,y_are_LinDep_&_not_y_is_zero_&_not_w_is_zero_)_)
assume  are_Prop u,v ; ::_thesis: ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
then A41: u,v,p are_LinDep by ANPROJ_1:11;
 ( q,r,q are_LinDep & p,p,q are_LinDep ) by ANPROJ_1:11;
hence  ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) by A3, A8, A41; ::_thesis: verum
end;
hence  ex y, w being Element of V st
( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero ) by A12; ::_thesis: verum
end;

theorem Th5: :: ANPROJ_2:5
for V being RealLinearSpace
for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds
for y being Element of V holds
( y is zero or not u,v,y are_LinDep or not u1,v1,y are_LinDep )
proof
let V be RealLinearSpace; ::_thesis: for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds
for y being Element of V holds
( y is zero or not u,v,y are_LinDep or not u1,v1,y are_LinDep )
let u, v, u1, v1 be Element of V; ::_thesis: ( ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) implies for y being Element of V holds
( y is zero or not u,v,y are_LinDep or not u1,v1,y are_LinDep ) )
assume A1: for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: for y being Element of V holds
( y is zero or not u,v,y are_LinDep or not u1,v1,y are_LinDep )
then A2: not are_Prop u,v by Th2;
assume  ex y being Element of V st
( not y is zero & u,v,y are_LinDep & u1,v1,y are_LinDep ) ; ::_thesis: contradiction
then consider y being Element of V such that
A3: not y is zero and
A4: u,v,y are_LinDep and
A5: u1,v1,y are_LinDep ;
 ( not u is zero & not v is zero ) by A1, Th2;
then consider a, b being Real such that
A6: y = (a * u) + (b * v) by A4, A2, ANPROJ_1:6;
A7: not are_Prop u1,v1 by A1, Th2;
 ( not u1 is zero & not v1 is zero ) by A1, Th2;
then consider a1, b1 being Real such that
A8: y = (a1 * u1) + (b1 * v1) by A5, A7, ANPROJ_1:6;
0. V = ((a * u) + (b * v)) - ((a1 * u1) + (b1 * v1)) by A6, A8, RLVECT_1:15
.= ((a * u) + (b * v)) + ((- 1) * ((a1 * u1) + (b1 * v1))) by RLVECT_1:16
.= ((a * u) + (b * v)) + (((- 1) * (a1 * u1)) + ((- 1) * (b1 * v1))) by RLVECT_1:def_5
.= ((a * u) + (b * v)) + ((((- 1) * a1) * u1) + ((- 1) * (b1 * v1))) by RLVECT_1:def_7
.= ((a * u) + (b * v)) + ((((- 1) * a1) * u1) + (((- 1) * b1) * v1)) by RLVECT_1:def_7
.= (((a * u) + (b * v)) + (((- 1) * a1) * u1)) + (((- 1) * b1) * v1) by RLVECT_1:def_3 ;
then  ( a = 0 & b = 0 ) by A1;
then y = (0. V) + (0 * v) by A6, RLVECT_1:10
.= (0. V) + (0. V) by RLVECT_1:10
.= 0. V by RLVECT_1:4 ;
hence  contradiction by A3; ::_thesis: verum
end;

definition
let V be RealLinearSpace;
let u, v, w be Element of V;
predu,v,w are_Prop_Vect  means :Def1: :: ANPROJ_2:def 1
( not u is zero & not v is zero & not w is zero );
end;

:: deftheorem Def1 defines are_Prop_Vect ANPROJ_2:def_1_:_
 for V being RealLinearSpace
for u, v, w being Element of V holds
( u,v,w are_Prop_Vect iff ( not u is zero & not v is zero & not w is zero ) );

definition
let V be RealLinearSpace;
let u, v, w, u1, v1, w1 be Element of V;
predu,v,w,u1,v1,w1 lie_on_a_triangle  means :Def2: :: ANPROJ_2:def 2
( u,v,w1 are_LinDep & u,w,v1 are_LinDep & v,w,u1 are_LinDep );
end;

:: deftheorem Def2 defines lie_on_a_triangle ANPROJ_2:def_2_:_
 for V being RealLinearSpace
for u, v, w, u1, v1, w1 being Element of V holds
( u,v,w,u1,v1,w1 lie_on_a_triangle iff ( u,v,w1 are_LinDep & u,w,v1 are_LinDep & v,w,u1 are_LinDep ) );

definition
let V be RealLinearSpace;
let o, u, v, w, u2, v2, w2 be Element of V;
predo,u,v,w,u2,v2,w2 are_perspective  means :Def3: :: ANPROJ_2:def 3
( o,u,u2 are_LinDep & o,v,v2 are_LinDep & o,w,w2 are_LinDep );
end;

:: deftheorem Def3 defines are_perspective ANPROJ_2:def_3_:_
 for V being RealLinearSpace
for o, u, v, w, u2, v2, w2 being Element of V holds
( o,u,v,w,u2,v2,w2 are_perspective iff ( o,u,u2 are_LinDep & o,v,v2 are_LinDep & o,w,w2 are_LinDep ) );

Lm8:  for V being RealLinearSpace
for o being Element of V
for a being Real holds - (a * o) = (- a) * o
proof
let V be RealLinearSpace; ::_thesis: for o being Element of V
for a being Real holds - (a * o) = (- a) * o
let o be Element of V; ::_thesis: for a being Real holds - (a * o) = (- a) * o
let a be Real; ::_thesis: - (a * o) = (- a) * o
thus  - (a * o) = a * (- o) by RLVECT_1:25
.= (- a) * o by RLVECT_1:24 ; ::_thesis: verum
end;

theorem Th6: :: ANPROJ_2:6
for V being RealLinearSpace
for o, u, u2 being Element of V st o,u,u2 are_LinDep & not are_Prop o,u & not are_Prop o,u2 & not are_Prop u,u2 & o,u,u2 are_Prop_Vect holds
( ex a1, b1 being Real st
( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) & ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) )
proof
let V be RealLinearSpace; ::_thesis: for o, u, u2 being Element of V st o,u,u2 are_LinDep & not are_Prop o,u & not are_Prop o,u2 & not are_Prop u,u2 & o,u,u2 are_Prop_Vect holds
( ex a1, b1 being Real st
( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) & ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) )
let o, u, u2 be Element of V; ::_thesis: ( o,u,u2 are_LinDep & not are_Prop o,u & not are_Prop o,u2 & not are_Prop u,u2 & o,u,u2 are_Prop_Vect implies ( ex a1, b1 being Real st
( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) & ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) ) )
assume that
A1: o,u,u2 are_LinDep and
A2: not are_Prop o,u and
A3: not are_Prop o,u2 and
A4: not are_Prop u,u2 and
A5: o,u,u2 are_Prop_Vect ; ::_thesis: ( ex a1, b1 being Real st
( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) & ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) )
consider a, b, c being Real such that
A6: ((a * o) + (b * u)) + (c * u2) = 0. V and
A7: ( a <> 0 or b <> 0 or c <> 0 ) by A1, ANPROJ_1:def_2;
 not u is zero by A5, Def1;
then A8: u <> 0. V ;
 not u2 is zero by A5, Def1;
then A9: u2 <> 0. V ;
 not o is zero by A5, Def1;
then A10: o <> 0. V ;
A11: ( a <> 0 & b <> 0 & c <> 0 )
proof
A12: now__::_thesis:_not_b_=_0
assume A13: b = 0 ; ::_thesis: contradiction
then  0. V = ((a * o) + (0. V)) + (c * u2) by A6, RLVECT_1:10
.= (a * o) + (c * u2) by RLVECT_1:4 ;
then a * o = - (c * u2) by RLVECT_1:6
.= c * (- u2) by RLVECT_1:25 ;
then A14: a * o = (- c) * u2 by RLVECT_1:24;
A15: ( a <> 0 & c <> 0 )
proof
A16: now__::_thesis:_not_c_=_0
assume A17: c = 0 ; ::_thesis: contradiction
then  0. V = ((a * o) + (0 * u)) + (0. V) by A6, A13, RLVECT_1:10
.= (a * o) + (0 * u) by RLVECT_1:4
.= (a * o) + (0. V) by RLVECT_1:10
.= a * o by RLVECT_1:4 ;
hence  contradiction by A7, A10, A13, A17, RLVECT_1:11; ::_thesis: verum
end;
A18: now__::_thesis:_not_a_=_0
assume A19: a = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (0 * u)) + (c * u2) by A6, A13, RLVECT_1:10
.= (0 * u) + (c * u2) by RLVECT_1:4
.= (0. V) + (c * u2) by RLVECT_1:10
.= c * u2 by RLVECT_1:4 ;
hence  contradiction by A7, A9, A13, A19, RLVECT_1:11; ::_thesis: verum
end;
assume  ( not a <> 0 or not c <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A18, A16; ::_thesis: verum
end;
then  - c <> 0 ;
hence  contradiction by A3, A15, A14, ANPROJ_1:def_1; ::_thesis: verum
end;
A20: now__::_thesis:_not_a_=_0
assume A21: a = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (b * u)) + (c * u2) by A6, RLVECT_1:10
.= (b * u) + (c * u2) by RLVECT_1:4 ;
then b * u = - (c * u2) by RLVECT_1:6
.= c * (- u2) by RLVECT_1:25 ;
then A22: b * u = (- c) * u2 by RLVECT_1:24;
A23: ( b <> 0 & c <> 0 )
proof
A24: now__::_thesis:_not_c_=_0
assume A25: c = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (b * u)) + (0 * u2) by A6, A21, RLVECT_1:10
.= (b * u) + (0 * u2) by RLVECT_1:4
.= (b * u) + (0. V) by RLVECT_1:10
.= b * u by RLVECT_1:4 ;
hence  contradiction by A7, A8, A21, A25, RLVECT_1:11; ::_thesis: verum
end;
A26: now__::_thesis:_not_b_=_0
assume A27: b = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (0 * u)) + (c * u2) by A6, A21, RLVECT_1:10
.= (0 * u) + (c * u2) by RLVECT_1:4
.= (0. V) + (c * u2) by RLVECT_1:10
.= c * u2 by RLVECT_1:4 ;
hence  contradiction by A7, A9, A21, A27, RLVECT_1:11; ::_thesis: verum
end;
assume  ( not b <> 0 or not c <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A26, A24; ::_thesis: verum
end;
then  - c <> 0 ;
hence  contradiction by A4, A23, A22, ANPROJ_1:def_1; ::_thesis: verum
end;
A28: now__::_thesis:_not_c_=_0
assume A29: c = 0 ; ::_thesis: contradiction
then  0. V = ((a * o) + (b * u)) + (0. V) by A6, RLVECT_1:10
.= (a * o) + (b * u) by RLVECT_1:4 ;
then a * o = - (b * u) by RLVECT_1:6
.= b * (- u) by RLVECT_1:25 ;
then A30: a * o = (- b) * u by RLVECT_1:24;
A31: ( a <> 0 & b <> 0 )
proof
A32: now__::_thesis:_not_b_=_0
assume A33: b = 0 ; ::_thesis: contradiction
then  0. V = ((a * o) + (0 * u)) + (0. V) by A6, A29, RLVECT_1:10
.= (a * o) + (0 * u) by RLVECT_1:4
.= (a * o) + (0. V) by RLVECT_1:10
.= a * o by RLVECT_1:4 ;
hence  contradiction by A7, A10, A29, A33, RLVECT_1:11; ::_thesis: verum
end;
A34: now__::_thesis:_not_a_=_0
assume A35: a = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (b * u)) + (0 * u2) by A6, A29, RLVECT_1:10
.= (b * u) + (0 * u2) by RLVECT_1:4
.= (b * u) + (0. V) by RLVECT_1:10
.= b * u by RLVECT_1:4 ;
hence  contradiction by A7, A8, A29, A35, RLVECT_1:11; ::_thesis: verum
end;
assume  ( not a <> 0 or not b <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A34, A32; ::_thesis: verum
end;
then  - b <> 0 ;
hence  contradiction by A2, A31, A30, ANPROJ_1:def_1; ::_thesis: verum
end;
assume  ( not a <> 0 or not b <> 0 or not c <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A20, A12, A28; ::_thesis: verum
end;
then A36: c " <> 0 by XCMPLX_1:202;
 a " <> 0 by A11, XCMPLX_1:202;
then A37: ( (a ") * b <> 0 & - ((a ") * c) <> 0 ) by A11, XCMPLX_1:6;
(a ") * (- (c * u2)) = (a ") * ((a * o) + (b * u)) by A6, RLVECT_1:6
.= ((a ") * (a * o)) + ((a ") * (b * u)) by RLVECT_1:def_5
.= (((a ") * a) * o) + ((a ") * (b * u)) by RLVECT_1:def_7
.= (((a ") * a) * o) + (((a ") * b) * u) by RLVECT_1:def_7
.= (1 * o) + (((a ") * b) * u) by A11, XCMPLX_0:def_7
.= o + (((a ") * b) * u) by RLVECT_1:def_8 ;
then o + (((a ") * b) * u) = (a ") * (c * (- u2)) by RLVECT_1:25
.= ((a ") * c) * (- u2) by RLVECT_1:def_7
.= (- ((a ") * c)) * u2 by RLVECT_1:24 ;
hence  ex a1, b1 being Real st
( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) by A37; ::_thesis: ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 )
 - b <> 0 by A11;
then A38: (c ") * (- b) <> 0 by A36, XCMPLX_1:6;
c * u2 = - ((a * o) + (b * u)) by A6, RLVECT_1:def_10
.= (- (a * o)) + (- (b * u)) by RLVECT_1:31
.= ((- a) * o) + (- (b * u)) by Lm8
.= ((- a) * o) + ((- b) * u) by Lm8 ;
then (c ") * (c * u2) = ((c ") * ((- a) * o)) + ((c ") * ((- b) * u)) by RLVECT_1:def_5
.= (((c ") * (- a)) * o) + ((c ") * ((- b) * u)) by RLVECT_1:def_7
.= (((c ") * (- a)) * o) + (((c ") * (- b)) * u) by RLVECT_1:def_7 ;
then A39: (((c ") * (- a)) * o) + (((c ") * (- b)) * u) = ((c ") * c) * u2 by RLVECT_1:def_7
.= 1 * u2 by A11, XCMPLX_0:def_7
.= u2 by RLVECT_1:def_8 ;
 - a <> 0 by A11;
then  (c ") * (- a) <> 0 by A36, XCMPLX_1:6;
hence  ex a2, c2 being Real st
( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) by A39, A38; ::_thesis: verum
end;

theorem Th7: :: ANPROJ_2:7
for V being RealLinearSpace
for p, q, r being Element of V st p,q,r are_LinDep & not are_Prop p,q & p,q,r are_Prop_Vect holds
ex a, b being Real st r = (a * p) + (b * q)
proof
let V be RealLinearSpace; ::_thesis: for p, q, r being Element of V st p,q,r are_LinDep & not are_Prop p,q & p,q,r are_Prop_Vect holds
ex a, b being Real st r = (a * p) + (b * q)
let p, q, r be Element of V; ::_thesis: ( p,q,r are_LinDep & not are_Prop p,q & p,q,r are_Prop_Vect implies ex a, b being Real st r = (a * p) + (b * q) )
assume that
A1: p,q,r are_LinDep and
A2: not are_Prop p,q and
A3: p,q,r are_Prop_Vect ; ::_thesis: ex a, b being Real st r = (a * p) + (b * q)
consider a, b, c being Real such that
A4: ((a * p) + (b * q)) + (c * r) = 0. V and
A5: ( a <> 0 or b <> 0 or c <> 0 ) by A1, ANPROJ_1:def_2;
 not q is zero by A3, Def1;
then A6: q <> 0. V ;
 not p is zero by A3, Def1;
then A7: p <> 0. V ;
A8: c <> 0
proof
assume A9: not c <> 0 ; ::_thesis: contradiction
then  0. V = ((a * p) + (b * q)) + (0. V) by A4, RLVECT_1:10
.= (a * p) + (b * q) by RLVECT_1:4 ;
then A10: a * p = - (b * q) by RLVECT_1:6
.= (- b) * q by Lm8 ;
A11: ( a <> 0 & b <> 0 )
proof
A12: now__::_thesis:_not_b_=_0
assume A13: b = 0 ; ::_thesis: contradiction
then  0. V = ((a * p) + (0. V)) + (0 * r) by A4, A9, RLVECT_1:10
.= ((a * p) + (0. V)) + (0. V) by RLVECT_1:10
.= (a * p) + (0. V) by RLVECT_1:4
.= a * p by RLVECT_1:4 ;
hence  contradiction by A7, A5, A9, A13, RLVECT_1:11; ::_thesis: verum
end;
A14: now__::_thesis:_not_a_=_0
assume A15: a = 0 ; ::_thesis: contradiction
then  0. V = ((0. V) + (b * q)) + (0 * r) by A4, A9, RLVECT_1:10
.= ((0. V) + (b * q)) + (0. V) by RLVECT_1:10
.= (b * q) + (0. V) by RLVECT_1:4
.= b * q by RLVECT_1:4 ;
hence  contradiction by A6, A5, A9, A15, RLVECT_1:11; ::_thesis: verum
end;
assume  ( not a <> 0 or not b <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A14, A12; ::_thesis: verum
end;
then  - b <> 0 ;
hence  contradiction by A2, A11, A10, ANPROJ_1:def_1; ::_thesis: verum
end;
c * r = - ((a * p) + (b * q)) by A4, RLVECT_1:def_10
.= (- (a * p)) + (- (b * q)) by RLVECT_1:31
.= ((- a) * p) + (- (b * q)) by Lm8
.= ((- a) * p) + ((- b) * q) by Lm8 ;
then (c ") * (c * r) = ((c ") * ((- a) * p)) + ((c ") * ((- b) * q)) by RLVECT_1:def_5
.= (((c ") * (- a)) * p) + ((c ") * ((- b) * q)) by RLVECT_1:def_7
.= (((c ") * (- a)) * p) + (((c ") * (- b)) * q) by RLVECT_1:def_7 ;
then (((c ") * (- a)) * p) + (((c ") * (- b)) * q) = ((c ") * c) * r by RLVECT_1:def_7
.= 1 * r by A8, XCMPLX_0:def_7
.= r by RLVECT_1:def_8 ;
hence  ex a, b being Real st r = (a * p) + (b * q) ; ::_thesis: verum
end;

Lm9:  for V being RealLinearSpace
for u2, w2 being Element of V
for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
proof
let V be RealLinearSpace; ::_thesis: for u2, w2 being Element of V
for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
let u2, w2 be Element of V; ::_thesis: for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
let b1 be Real; ::_thesis: ( b1 * u2 = w2 & b1 <> 0 implies are_Prop u2,w2 )
assume that
A1: b1 * u2 = w2 and
A2: b1 <> 0 ; ::_thesis: are_Prop u2,w2
 b1 * u2 = 1 * w2 by A1, RLVECT_1:def_8;
hence  are_Prop u2,w2 by A2, ANPROJ_1:def_1; ::_thesis: verum
end;

Lm10:  for V being RealLinearSpace
for q, o, p, r, s being Element of V
for a, b being Real st q = o + (a * p) & r = o + (b * s) & are_Prop q,r & a <> 0 holds
o,p,s are_LinDep
proof
let V be RealLinearSpace; ::_thesis: for q, o, p, r, s being Element of V
for a, b being Real st q = o + (a * p) & r = o + (b * s) & are_Prop q,r & a <> 0 holds
o,p,s are_LinDep
let q, o, p, r, s be Element of V; ::_thesis: for a, b being Real st q = o + (a * p) & r = o + (b * s) & are_Prop q,r & a <> 0 holds
o,p,s are_LinDep
let a, b be Real; ::_thesis: ( q = o + (a * p) & r = o + (b * s) & are_Prop q,r & a <> 0 implies o,p,s are_LinDep )
assume that
A1: ( q = o + (a * p) & r = o + (b * s) & are_Prop q,r ) and
A2: a <> 0 ; ::_thesis: o,p,s are_LinDep
consider A being Real such that
 A <> 0 and
A3: o + (a * p) = A * (o + (b * s)) by A1, ANPROJ_1:1;
o + (a * p) = (A * o) + (A * (b * s)) by A3, RLVECT_1:def_5
.= (A * o) + ((A * b) * s) by RLVECT_1:def_7 ;
then (- (A * o)) + (o + (a * p)) = ((- (A * o)) + (A * o)) + ((A * b) * s) by RLVECT_1:def_3
.= (0. V) + ((A * b) * s) by RLVECT_1:5
.= (A * b) * s by RLVECT_1:4 ;
then  ((- (A * o)) + o) + (a * p) = (A * b) * s by RLVECT_1:def_3;
then (A * b) * s = (((- A) * o) + o) + (a * p) by Lm8
.= (((- A) * o) + (1 * o)) + (a * p) by RLVECT_1:def_8
.= (((- A) + 1) * o) + (a * p) by RLVECT_1:def_6 ;
then  ((((- A) + 1) * o) + (a * p)) + (- ((A * b) * s)) = 0. V by RLVECT_1:5;
then  0. V = ((((- A) + 1) * o) + (a * p)) + ((- (A * b)) * s) by Lm8;
hence  o,p,s are_LinDep by A2, ANPROJ_1:def_2; ::_thesis: verum
end;

Lm11:  for V being RealLinearSpace
for p, q being Element of V
for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
proof
let V be RealLinearSpace; ::_thesis: for p, q being Element of V
for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
let p, q be Element of V; ::_thesis: for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
let a be Real; ::_thesis: ( a * p = q & a <> 0 & not p is zero implies not q is zero )
assume that
A1: ( a * p = q & a <> 0 ) and
A2: not p is zero ; ::_thesis: not q is zero
 p <> 0. V by A2;
then  q <> 0. V by A1, RLVECT_1:11;
hence  not q is zero by STRUCT_0:def_12; ::_thesis: verum
end;

Lm12:  for V being RealLinearSpace
for r, u2, v2, o, u, v being Element of V
for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
proof
let V be RealLinearSpace; ::_thesis: for r, u2, v2, o, u, v being Element of V
for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let r, u2, v2, o, u, v be Element of V; ::_thesis: for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let a1, a2 be Real; ::_thesis: for A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let A, B be Real; ::_thesis: ( r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) implies r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v) )
assume  ( r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) ) ; ::_thesis: r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
hence r = ((A * o) + (A * (a1 * u))) + (B * (o + (a2 * v))) by RLVECT_1:def_5
.= ((A * o) + (A * (a1 * u))) + ((B * o) + (B * (a2 * v))) by RLVECT_1:def_5
.= ((A * o) + ((A * a1) * u)) + ((B * o) + (B * (a2 * v))) by RLVECT_1:def_7
.= ((A * o) + ((A * a1) * u)) + ((B * o) + ((B * a2) * v)) by RLVECT_1:def_7
.= (((A * o) + ((A * a1) * u)) + (B * o)) + ((B * a2) * v) by RLVECT_1:def_3
.= (((A * o) + (B * o)) + ((A * a1) * u)) + ((B * a2) * v) by RLVECT_1:def_3
.= (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v) by RLVECT_1:def_6 ;
::_thesis: verum
end;

Lm13:  for V being RealLinearSpace
for r, p, q, o being Element of V
for a, b being Real st r = (a * p) + (b * q) holds
r = ((0 * o) + (a * p)) + (b * q)
proof
let V be RealLinearSpace; ::_thesis: for r, p, q, o being Element of V
for a, b being Real st r = (a * p) + (b * q) holds
r = ((0 * o) + (a * p)) + (b * q)
let r, p, q, o be Element of V; ::_thesis: for a, b being Real st r = (a * p) + (b * q) holds
r = ((0 * o) + (a * p)) + (b * q)
let a, b be Real; ::_thesis: ( r = (a * p) + (b * q) implies r = ((0 * o) + (a * p)) + (b * q) )
assume  r = (a * p) + (b * q) ; ::_thesis: r = ((0 * o) + (a * p)) + (b * q)
hence r = ((0. V) + (a * p)) + (b * q) by RLVECT_1:4
.= ((0 * o) + (a * p)) + (b * q) by RLVECT_1:10 ;
::_thesis: verum
end;

Lm14:  for V being RealLinearSpace
for p, q being Element of V holds (0 * p) + (0 * q) = 0. V
proof
let V be RealLinearSpace; ::_thesis: for p, q being Element of V holds (0 * p) + (0 * q) = 0. V
let p, q be Element of V; ::_thesis: (0 * p) + (0 * q) = 0. V
thus (0 * p) + (0 * q) = (0. V) + (0 * q) by RLVECT_1:10
.= 0 * q by RLVECT_1:4
.= 0. V by RLVECT_1:10 ; ::_thesis: verum
end;

Lm15:  for V being RealLinearSpace
for o, v, w being Element of V
for b, a2, a3 being Real holds ((0 * o) + ((b * a2) * v)) + (((- b) * a3) * w) = b * ((a2 * v) - (a3 * w))
proof
let V be RealLinearSpace; ::_thesis: for o, v, w being Element of V
for b, a2, a3 being Real holds ((0 * o) + ((b * a2) * v)) + (((- b) * a3) * w) = b * ((a2 * v) - (a3 * w))
let o, v, w be Element of V; ::_thesis: for b, a2, a3 being Real holds ((0 * o) + ((b * a2) * v)) + (((- b) * a3) * w) = b * ((a2 * v) - (a3 * w))
let b, a2, a3 be Real; ::_thesis: ((0 * o) + ((b * a2) * v)) + (((- b) * a3) * w) = b * ((a2 * v) - (a3 * w))
thus ((0 * o) + ((b * a2) * v)) + (((- b) * a3) * w) = ((0. V) + ((b * a2) * v)) + (((- b) * a3) * w) by RLVECT_1:10
.= ((b * a2) * v) + (((- b) * a3) * w) by RLVECT_1:4
.= (b * (a2 * v)) + ((b * (- a3)) * w) by RLVECT_1:def_7
.= (b * (a2 * v)) + (b * ((- a3) * w)) by RLVECT_1:def_7
.= (b * (a2 * v)) + (b * (- (a3 * w))) by Lm8
.= b * ((a2 * v) - (a3 * w)) by RLVECT_1:def_5 ; ::_thesis: verum
end;

theorem Th8: :: ANPROJ_2:8
for V being RealLinearSpace
for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 are_perspective & not are_Prop o,u2 & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,u2 & not are_Prop v,v2 & not are_Prop w,w2 & not o,u,v are_LinDep & not o,u,w are_LinDep & not o,v,w are_LinDep & u,v,w,u1,v1,w1 lie_on_a_triangle & u2,v2,w2,u1,v1,w1 lie_on_a_triangle holds
u1,v1,w1 are_LinDep
proof
let V be RealLinearSpace; ::_thesis: for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 are_perspective & not are_Prop o,u2 & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,u2 & not are_Prop v,v2 & not are_Prop w,w2 & not o,u,v are_LinDep & not o,u,w are_LinDep & not o,v,w are_LinDep & u,v,w,u1,v1,w1 lie_on_a_triangle & u2,v2,w2,u1,v1,w1 lie_on_a_triangle holds
u1,v1,w1 are_LinDep
let o, u, v, w, u2, v2, w2, u1, v1, w1 be Element of V; ::_thesis: ( not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 are_perspective & not are_Prop o,u2 & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,u2 & not are_Prop v,v2 & not are_Prop w,w2 & not o,u,v are_LinDep & not o,u,w are_LinDep & not o,v,w are_LinDep & u,v,w,u1,v1,w1 lie_on_a_triangle & u2,v2,w2,u1,v1,w1 lie_on_a_triangle implies u1,v1,w1 are_LinDep )
assume that
A1: not o is zero and
A2: u,v,w are_Prop_Vect and
A3: u2,v2,w2 are_Prop_Vect and
A4: u1,v1,w1 are_Prop_Vect and
A5: o,u,v,w,u2,v2,w2 are_perspective and
A6: not are_Prop o,u2 and
A7: not are_Prop o,v2 and
A8: not are_Prop o,w2 and
A9: not are_Prop u,u2 and
A10: not are_Prop v,v2 and
A11: not are_Prop w,w2 and
A12: not o,u,v are_LinDep and
A13: not o,u,w are_LinDep and
A14: not o,v,w are_LinDep and
A15: u,v,w,u1,v1,w1 lie_on_a_triangle and
A16: u2,v2,w2,u1,v1,w1 lie_on_a_triangle ; ::_thesis: u1,v1,w1 are_LinDep
A17: not w is zero by A2, Def1;
A18: ( o,w,w2 are_LinDep & not are_Prop w,o ) by A5, A13, Def3, ANPROJ_1:11;
A19: not w2 is zero by A3, Def1;
then  o,w,w2 are_Prop_Vect by A1, A17, Def1;
then consider a3, b3 being Real such that
A20: b3 * w2 = o + (a3 * w) and
 a3 <> 0 and
A21: b3 <> 0 by A8, A11, A18, Th6;
A22: not u is zero by A2, Def1;
A23: not v is zero by A2, Def1;
A24: ( o,v,v2 are_LinDep & not are_Prop o,v ) by A5, A12, Def3, ANPROJ_1:11;
A25: ( o,u,u2 are_LinDep & not are_Prop o,u ) by A5, A12, Def3, ANPROJ_1:11;
A26: not u2 is zero by A3, Def1;
then  o,u,u2 are_Prop_Vect by A1, A22, Def1;
then consider a1, b1 being Real such that
A27: b1 * u2 = o + (a1 * u) and
A28: a1 <> 0 and
A29: b1 <> 0 by A6, A9, A25, Th6;
A30: not v2 is zero by A3, Def1;
then  o,v,v2 are_Prop_Vect by A1, A23, Def1;
then consider a2, b2 being Real such that
A31: b2 * v2 = o + (a2 * v) and
A32: a2 <> 0 and
A33: b2 <> 0 by A7, A10, A24, Th6;
set u29 = o + (a1 * u);
set v29 = o + (a2 * v);
set w29 = o + (a3 * w);
A34: are_Prop v2,o + (a2 * v) by A31, A33, Lm9;
A35: not o + (a2 * v) is zero by A30, A31, A33, Lm11;
A36: are_Prop w2,o + (a3 * w) by A20, A21, Lm9;
A37: ( u,v,w1 are_LinDep & not are_Prop u,v ) by A12, A15, Def2, ANPROJ_1:12;
A38: not w1 is zero by A4, Def1;
then  u,v,w1 are_Prop_Vect by A22, A23, Def1;
then consider c3, d3 being Real such that
A39: w1 = (c3 * u) + (d3 * v) by A37, Th7;
A40: are_Prop u2,o + (a1 * u) by A27, A29, Lm9;
A41: ( v,w,u1 are_LinDep & not are_Prop v,w ) by A14, A15, Def2, ANPROJ_1:12;
A42: not u1 is zero by A4, Def1;
then  v,w,u1 are_Prop_Vect by A23, A17, Def1;
then consider c1, d1 being Real such that
A43: u1 = (c1 * v) + (d1 * w) by A41, Th7;
 v2,w2,u1 are_LinDep by A16, Def2;
then A44: o + (a2 * v),o + (a3 * w),u1 are_LinDep by A34, A36, ANPROJ_1:4;
A45: not are_Prop o + (a2 * v),o + (a3 * w) by A14, A32, Lm10;
A46: not o + (a3 * w) is zero by A19, A20, A21, Lm11;
then  o + (a2 * v),o + (a3 * w),u1 are_Prop_Vect by A42, A35, Def1;
then consider A1, B1 being Real such that
A47: u1 = (A1 * (o + (a2 * v))) + (B1 * (o + (a3 * w))) by A44, A45, Th7;
A48: ( u,w,v1 are_LinDep & not are_Prop u,w ) by A13, A15, Def2, ANPROJ_1:12;
A49: not v1 is zero by A4, Def1;
then  u,w,v1 are_Prop_Vect by A22, A17, Def1;
then consider c2, d2 being Real such that
A50: v1 = (c2 * u) + (d2 * w) by A48, Th7;
A51: u1 = (((A1 + B1) * o) + ((A1 * a2) * v)) + ((B1 * a3) * w) by A47, Lm12;
 u2,v2,w1 are_LinDep by A16, Def2;
then A52: o + (a1 * u),o + (a2 * v),w1 are_LinDep by A40, A34, ANPROJ_1:4;
A53: not are_Prop o + (a1 * u),o + (a2 * v) by A12, A28, Lm10;
A54: not o + (a1 * u) is zero by A26, A27, A29, Lm11;
then  o + (a1 * u),o + (a2 * v),w1 are_Prop_Vect by A38, A35, Def1;
then consider A3, B3 being Real such that
A55: w1 = (A3 * (o + (a1 * u))) + (B3 * (o + (a2 * v))) by A52, A53, Th7;
 u2,w2,v1 are_LinDep by A16, Def2;
then A56: o + (a1 * u),o + (a3 * w),v1 are_LinDep by A40, A36, ANPROJ_1:4;
A57: not are_Prop o + (a1 * u),o + (a3 * w) by A13, A28, Lm10;
A58: w1 = (((A3 + B3) * o) + ((A3 * a1) * u)) + ((B3 * a2) * v) by A55, Lm12;
 o + (a1 * u),o + (a3 * w),v1 are_Prop_Vect by A49, A54, A46, Def1;
then consider A2, B2 being Real such that
A59: v1 = (A2 * (o + (a1 * u))) + (B2 * (o + (a3 * w))) by A56, A57, Th7;
A60: v1 = (((A2 + B2) * o) + ((A2 * a1) * u)) + ((B2 * a3) * w) by A59, Lm12;
 w1 = ((0 * o) + (c3 * u)) + (d3 * v) by A39, Lm13;
then A61: A3 + B3 = 0 by A12, A58, ANPROJ_1:8;
 u1 = ((0 * o) + (c1 * v)) + (d1 * w) by A43, Lm13;
then A62: A1 + B1 = 0 by A14, A51, ANPROJ_1:8;
 v1 = ((0 * o) + (c2 * u)) + (d2 * w) by A50, Lm13;
then A63: A2 + B2 = 0 by A13, A60, ANPROJ_1:8;
A64: ( A1 <> 0 & A2 <> 0 & A3 <> 0 )
proof
A65: now__::_thesis:_not_A2_=_0
assume  A2 = 0 ; ::_thesis: contradiction
then  v1 = 0. V by A59, A63, Lm14;
hence  contradiction by A49; ::_thesis: verum
end;
A66: now__::_thesis:_not_A1_=_0
assume  A1 = 0 ; ::_thesis: contradiction
then  u1 = 0. V by A47, A62, Lm14;
hence  contradiction by A42; ::_thesis: verum
end;
A67: now__::_thesis:_not_A3_=_0
assume  A3 = 0 ; ::_thesis: contradiction
then  w1 = 0. V by A55, A61, Lm14;
hence  contradiction by A38; ::_thesis: verum
end;
assume  ( not A1 <> 0 or not A2 <> 0 or not A3 <> 0 ) ; ::_thesis: contradiction
hence  contradiction by A66, A65, A67; ::_thesis: verum
end;
set u19 = (a2 * v) - (a3 * w);
set v19 = (a1 * u) - (a3 * w);
set w19 = (a1 * u) - (a2 * v);
 B1 = - A1 by A62;
then  u1 = A1 * ((a2 * v) - (a3 * w)) by A51, Lm15;
then A68: are_Prop (a2 * v) - (a3 * w),u1 by A64, Lm9;
 B3 = - A3 by A61;
then  w1 = A3 * ((a1 * u) - (a2 * v)) by A58, Lm15;
then A69: are_Prop (a1 * u) - (a2 * v),w1 by A64, Lm9;
 B2 = - A2 by A63;
then  v1 = A2 * ((a1 * u) - (a3 * w)) by A60, Lm15;
then A70: are_Prop (a1 * u) - (a3 * w),v1 by A64, Lm9;
((1 * ((a2 * v) - (a3 * w))) + ((- 1) * ((a1 * u) - (a3 * w)))) + (1 * ((a1 * u) - (a2 * v))) = (((a2 * v) - (a3 * w)) + ((- 1) * ((a1 * u) - (a3 * w)))) + (1 * ((a1 * u) - (a2 * v))) by RLVECT_1:def_8
.= (((a2 * v) - (a3 * w)) + ((- 1) * ((a1 * u) - (a3 * w)))) + ((a1 * u) - (a2 * v)) by RLVECT_1:def_8
.= (((a2 * v) - (a3 * w)) + (- ((a1 * u) - (a3 * w)))) + ((a1 * u) - (a2 * v)) by RLVECT_1:16
.= (((a2 * v) + (- (a3 * w))) + ((a3 * w) + (- (a1 * u)))) + ((a1 * u) - (a2 * v)) by RLVECT_1:33
.= ((((a2 * v) + (- (a3 * w))) + (a3 * w)) + (- (a1 * u))) + ((a1 * u) + (- (a2 * v))) by RLVECT_1:def_3
.= (((a2 * v) + ((- (a3 * w)) + (a3 * w))) + (- (a1 * u))) + ((a1 * u) + (- (a2 * v))) by RLVECT_1:def_3
.= (((a2 * v) + (0. V)) + (- (a1 * u))) + ((a1 * u) + (- (a2 * v))) by RLVECT_1:5
.= ((a2 * v) + (- (a1 * u))) + ((a1 * u) + (- (a2 * v))) by RLVECT_1:4
.= (a2 * v) + ((- (a1 * u)) + ((a1 * u) + (- (a2 * v)))) by RLVECT_1:def_3
.= (a2 * v) + (((- (a1 * u)) + (a1 * u)) + (- (a2 * v))) by RLVECT_1:def_3
.= (a2 * v) + ((0. V) + (- (a2 * v))) by RLVECT_1:5
.= (a2 * v) + (- (a2 * v)) by RLVECT_1:4
.= 0. V by RLVECT_1:5 ;
then  (a2 * v) - (a3 * w),(a1 * u) - (a3 * w),(a1 * u) - (a2 * v) are_LinDep by ANPROJ_1:def_2;
hence  u1,v1,w1 are_LinDep by A68, A70, A69, ANPROJ_1:4; ::_thesis: verum
end;

definition
let V be RealLinearSpace;
let o, u, v, w, u2, v2, w2 be Element of V;
predo,u,v,w,u2,v2,w2 lie_on_an_angle  means :Def4: :: ANPROJ_2:def 4
( not o,u,u2 are_LinDep & o,u,v are_LinDep & o,u,w are_LinDep & o,u2,v2 are_LinDep & o,u2,w2 are_LinDep );
end;

:: deftheorem Def4 defines lie_on_an_angle ANPROJ_2:def_4_:_
 for V being RealLinearSpace
for o, u, v, w, u2, v2, w2 being Element of V holds
( o,u,v,w,u2,v2,w2 lie_on_an_angle iff ( not o,u,u2 are_LinDep & o,u,v are_LinDep & o,u,w are_LinDep & o,u2,v2 are_LinDep & o,u2,w2 are_LinDep ) );

definition
let V be RealLinearSpace;
let o, u, v, w, u2, v2, w2 be Element of V;
predo,u,v,w,u2,v2,w2 are_half_mutually_not_Prop  means :Def5: :: ANPROJ_2:def 5
( not are_Prop o,v & not are_Prop o,w & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,v & not are_Prop u,w & not are_Prop u2,v2 & not are_Prop u2,w2 & not are_Prop v,w & not are_Prop v2,w2 );
end;

:: deftheorem Def5 defines are_half_mutually_not_Prop ANPROJ_2:def_5_:_
 for V being RealLinearSpace
for o, u, v, w, u2, v2, w2 being Element of V holds
( o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop iff ( not are_Prop o,v & not are_Prop o,w & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,v & not are_Prop u,w & not are_Prop u2,v2 & not are_Prop u2,w2 & not are_Prop v,w & not are_Prop v2,w2 ) );

Lm16:  for V being RealLinearSpace
for u2, w2 being Element of V
for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
proof
let V be RealLinearSpace; ::_thesis: for u2, w2 being Element of V
for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
let u2, w2 be Element of V; ::_thesis: for b1 being Real st b1 * u2 = w2 & b1 <> 0 holds
are_Prop u2,w2
let b1 be Real; ::_thesis: ( b1 * u2 = w2 & b1 <> 0 implies are_Prop u2,w2 )
assume that
A1: b1 * u2 = w2 and
A2: b1 <> 0 ; ::_thesis: are_Prop u2,w2
 b1 * u2 = 1 * w2 by A1, RLVECT_1:def_8;
hence  are_Prop u2,w2 by A2, ANPROJ_1:def_1; ::_thesis: verum
end;

Lm17:  for V being RealLinearSpace
for p, q, y being Element of V
for a being Real st not are_Prop p,q & y = a * q & a <> 0 holds
not are_Prop p,y
proof
let V be RealLinearSpace; ::_thesis: for p, q, y being Element of V
for a being Real st not are_Prop p,q & y = a * q & a <> 0 holds
not are_Prop p,y
let p, q, y be Element of V; ::_thesis: for a being Real st not are_Prop p,q & y = a * q & a <> 0 holds
not are_Prop p,y
let a be Real; ::_thesis: ( not are_Prop p,q & y = a * q & a <> 0 implies not are_Prop p,y )
assume that
A1: not are_Prop p,q and
A2: ( y = a * q & a <> 0 ) ; ::_thesis: not are_Prop p,y
assume  are_Prop p,y ; ::_thesis: contradiction
then consider b being Real such that
A3: ( b <> 0 & p = b * y ) by ANPROJ_1:1;
 ( p = (b * a) * q & b * a <> 0 ) by A2, A3, RLVECT_1:def_7, XCMPLX_1:6;
hence  contradiction by A1, ANPROJ_1:1; ::_thesis: verum
end;

Lm18:  for V being RealLinearSpace
for w1, u, v2, o, u2 being Element of V
for a, b, c2 being Real st w1 = (a * u) + (b * v2) & v2 = o + (c2 * u2) holds
w1 = ((b * o) + (a * u)) + ((b * c2) * u2)
proof
let V be RealLinearSpace; ::_thesis: for w1, u, v2, o, u2 being Element of V
for a, b, c2 being Real st w1 = (a * u) + (b * v2) & v2 = o + (c2 * u2) holds
w1 = ((b * o) + (a * u)) + ((b * c2) * u2)
let w1, u, v2, o, u2 be Element of V; ::_thesis: for a, b, c2 being Real st w1 = (a * u) + (b * v2) & v2 = o + (c2 * u2) holds
w1 = ((b * o) + (a * u)) + ((b * c2) * u2)
let a, b, c2 be Real; ::_thesis: ( w1 = (a * u) + (b * v2) & v2 = o + (c2 * u2) implies w1 = ((b * o) + (a * u)) + ((b * c2) * u2) )
assume  ( w1 = (a * u) + (b * v2) & v2 = o + (c2 * u2) ) ; ::_thesis: w1 = ((b * o) + (a * u)) + ((b * c2) * u2)
hence w1 = (a * u) + ((b * o) + (b * (c2 * u2))) by RLVECT_1:def_5
.= ((a * u) + (b * o)) + (b * (c2 * u2)) by RLVECT_1:def_3
.= ((b * o) + (a * u)) + ((b * c2) * u2) by RLVECT_1:def_7 ;
::_thesis: verum
end;

Lm19:  for V being RealLinearSpace
for w1, u2, v1, o, u being Element of V
for a, b, a2 being Real st w1 = (a * u2) + (b * v1) & v1 = o + (a2 * u) holds
w1 = ((b * o) + ((b * a2) * u)) + (a * u2)
proof
let V be RealLinearSpace; ::_thesis: for w1, u2, v1, o, u being Element of V
for a, b, a2 being Real st w1 = (a * u2) + (b * v1) & v1 = o + (a2 * u) holds
w1 = ((b * o) + ((b * a2) * u)) + (a * u2)
let w1, u2, v1, o, u be Element of V; ::_thesis: for a, b, a2 being Real st w1 = (a * u2) + (b * v1) & v1 = o + (a2 * u) holds
w1 = ((b * o) + ((b * a2) * u)) + (a * u2)
let a, b, a2 be Real; ::_thesis: ( w1 = (a * u2) + (b * v1) & v1 = o + (a2 * u) implies w1 = ((b * o) + ((b * a2) * u)) + (a * u2) )
assume  ( w1 = (a * u2) + (b * v1) & v1 = o + (a2 * u) ) ; ::_thesis: w1 = ((b * o) + ((b * a2) * u)) + (a * u2)
hence w1 = ((b * o) + (a * u2)) + ((b * a2) * u) by Lm18
.= ((b * o) + ((b * a2) * u)) + (a * u2) by RLVECT_1:def_3 ;
::_thesis: verum
end;

Lm20:  for V being RealLinearSpace
for p, q being Element of V
for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
proof
let V be RealLinearSpace; ::_thesis: for p, q being Element of V
for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
let p, q be Element of V; ::_thesis: for a being Real st a * p = q & a <> 0 & not p is zero holds
not q is zero
let a be Real; ::_thesis: ( a * p = q & a <> 0 & not p is zero implies not q is zero )
assume that
A1: ( a * p = q & a <> 0 ) and
A2: not p is zero ; ::_thesis: not q is zero
 p <> 0. V by A2;
then  q <> 0. V by A1, RLVECT_1:11;
hence  not q is zero by STRUCT_0:def_12; ::_thesis: verum
end;

Lm21:  for V being RealLinearSpace
for p, q, y, s being Element of V
for a, b being Real st not are_Prop p,q & y = a * q & a <> 0 & s = b * p & b <> 0 holds
not are_Prop s,y
proof
let V be RealLinearSpace; ::_thesis: for p, q, y, s being Element of V
for a, b being Real st not are_Prop p,q & y = a * q & a <> 0 & s = b * p & b <> 0 holds
not are_Prop s,y
let p, q, y, s be Element of V; ::_thesis: for a, b being Real st not are_Prop p,q & y = a * q & a <> 0 & s = b * p & b <> 0 holds
not are_Prop s,y
let a, b be Real; ::_thesis: ( not are_Prop p,q & y = a * q & a <> 0 & s = b * p & b <> 0 implies not are_Prop s,y )
assume that
A1: not are_Prop p,q and
A2: ( y = a * q & a <> 0 ) and
A3: ( s = b * p & b <> 0 ) ; ::_thesis: not are_Prop s,y
assume  are_Prop s,y ; ::_thesis: contradiction
then consider c being Real such that
A4: ( c <> 0 & s = c * y ) by ANPROJ_1:1;
 ( s = (c * a) * q & c * a <> 0 ) by A2, A4, RLVECT_1:def_7, XCMPLX_1:6;
hence  contradiction by A1, A3, ANPROJ_1:def_1; ::_thesis: verum
end;

Lm22:  for V being RealLinearSpace
for r, u2, v2, o, u, v being Element of V
for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
proof
let V be RealLinearSpace; ::_thesis: for r, u2, v2, o, u, v being Element of V
for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let r, u2, v2, o, u, v be Element of V; ::_thesis: for a1, a2, A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let a1, a2 be Real; ::_thesis: for A, B being Real st r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) holds
r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
let A, B be Real; ::_thesis: ( r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) implies r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v) )
assume  ( r = (A * u2) + (B * v2) & u2 = o + (a1 * u) & v2 = o + (a2 * v) ) ; ::_thesis: r = (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v)
hence r = ((A * o) + (A * (a1 * u))) + (B * (o + (a2 * v))) by RLVECT_1:def_5
.= ((A * o) + (A * (a1 * u))) + ((B * o) + (B * (a2 * v))) by RLVECT_1:def_5
.= ((A * o) + ((A * a1) * u)) + ((B * o) + (B * (a2 * v))) by RLVECT_1:def_7
.= ((A * o) + ((A * a1) * u)) + ((B * o) + ((B * a2) * v)) by RLVECT_1:def_7
.= (((A * o) + ((A * a1) * u)) + (B * o)) + ((B * a2) * v) by RLVECT_1:def_3
.= (((A * o) + (B * o)) + ((A * a1) * u)) + ((B * a2) * v) by RLVECT_1:def_3
.= (((A + B) * o) + ((A * a1) * u)) + ((B * a2) * v) by RLVECT_1:def_6 ;
::_thesis: verum
end;

Lm23:  for a2, a3, c2 being Real st a2 <> a3 & c2 <> 0 holds
(a3 * c2) - (a2 * c2) <> 0
proof
let a2, a3, c2 be Real; ::_thesis: ( a2 <> a3 & c2 <> 0 implies (a3 * c2) - (a2 * c2) <> 0 )
assume that
A1: a2 <> a3 and
A2: c2 <> 0 ; ::_thesis: (a3 * c2) - (a2 * c2) <> 0
 ( (a3 * c2) - (a2 * c2) = (a3 - a2) * c2 & a3 - a2 <> 0 ) by A1;
hence  (a3 * c2) - (a2 * c2) <> 0 by A2, XCMPLX_1:6; ::_thesis: verum
end;

Lm24:  for a2, a3, c3, c2, A1, A19, B1, B19 being Real st A1 + B1 = A19 + B19 & A1 * a2 = A19 * a3 & B1 * c3 = B19 * c2 & a2 <> a3 & c2 <> 0 holds
A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1
proof
let a2, a3, c3, c2 be Real; ::_thesis: for A1, A19, B1, B19 being Real st A1 + B1 = A19 + B19 & A1 * a2 = A19 * a3 & B1 * c3 = B19 * c2 & a2 <> a3 & c2 <> 0 holds
A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1
let A1, A19, B1, B19 be Real; ::_thesis: ( A1 + B1 = A19 + B19 & A1 * a2 = A19 * a3 & B1 * c3 = B19 * c2 & a2 <> a3 & c2 <> 0 implies A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 )
assume that
A1: A1 + B1 = A19 + B19 and
A2: ( A1 * a2 = A19 * a3 & B1 * c3 = B19 * c2 ) and
A3: ( a2 <> a3 & c2 <> 0 ) ; ::_thesis: A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1
A4: (A1 * (a3 * c2)) + (B1 * (a3 * c2)) = (A19 + B19) * (a3 * c2) by A1, XCMPLX_1:8;
 ( A1 * (a2 * c2) = (A19 * a3) * c2 & B1 * (c3 * a3) = (B19 * c2) * a3 ) by A2, XCMPLX_1:4;
then  B1 * ((a3 * c3) - (a3 * c2)) = A1 * ((a3 * c2) - (a2 * c2)) by A4;
then A5: A1 * (((a3 * c2) - (a2 * c2)) * (((a3 * c2) - (a2 * c2)) ")) = (B1 * ((a3 * c3) - (a3 * c2))) * (((a3 * c2) - (a2 * c2)) ") by XCMPLX_1:4;
 (a3 * c2) - (a2 * c2) <> 0 by A3, Lm23;
then  A1 * 1 = (B1 * ((a3 * c3) - (a3 * c2))) * (((a3 * c2) - (a2 * c2)) ") by A5, XCMPLX_0:def_7;
hence  A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 ; ::_thesis: verum
end;

Lm25:  for c2, a2, a3, B1 being Real st c2 <> 0 & a2 <> a3 & B1 <> 0 holds
B1 * (((a3 * c2) - (a2 * c2)) ") <> 0
proof
let c2, a2, a3, B1 be Real; ::_thesis: ( c2 <> 0 & a2 <> a3 & B1 <> 0 implies B1 * (((a3 * c2) - (a2 * c2)) ") <> 0 )
assume that
A1: ( c2 <> 0 & a2 <> a3 ) and
A2: B1 <> 0 ; ::_thesis: B1 * (((a3 * c2) - (a2 * c2)) ") <> 0
 ((a3 * c2) - (a2 * c2)) " <> 0 by A1, Lm23, XCMPLX_1:202;
hence  B1 * (((a3 * c2) - (a2 * c2)) ") <> 0 by A2, XCMPLX_1:6; ::_thesis: verum
end;

Lm26:  for V being RealLinearSpace
for u1, o, u, u2 being Element of V
for a3, c3, c2, a2, A1, B1 being Real st A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 & c2 <> 0 & a2 <> a3 & u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) holds
u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))
proof
let V be RealLinearSpace; ::_thesis: for u1, o, u, u2 being Element of V
for a3, c3, c2, a2, A1, B1 being Real st A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 & c2 <> 0 & a2 <> a3 & u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) holds
u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))
let u1, o, u, u2 be Element of V; ::_thesis: for a3, c3, c2, a2, A1, B1 being Real st A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 & c2 <> 0 & a2 <> a3 & u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) holds
u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))
let a3, c3, c2, a2 be Real; ::_thesis: for A1, B1 being Real st A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 & c2 <> 0 & a2 <> a3 & u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) holds
u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))
let A1, B1 be Real; ::_thesis: ( A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 & c2 <> 0 & a2 <> a3 & u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) implies u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2)) )
assume that
A1: A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 and
A2: ( c2 <> 0 & a2 <> a3 ) and
A3: u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) ; ::_thesis: u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))
A4: (a3 * c2) - (a2 * c2) <> 0 by A2, Lm23;
A5: (B1 * c3) * u2 = ((B1 * 1) * c3) * u2
.= ((B1 * ((((a3 * c2) - (a2 * c2)) ") * ((a3 * c2) - (a2 * c2)))) * c3) * u2 by A4, XCMPLX_0:def_7
.= ((B1 * (((a3 * c2) - (a2 * c2)) ")) * ((c3 * c2) * (a3 - a2))) * u2
.= (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((c2 * c3) * (a3 - a2)) * u2) by RLVECT_1:def_7 ;
A6: (((((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1) * a2) * u = ((B1 * (((a3 * c2) - (a2 * c2)) ")) * ((a2 * a3) * (c3 - c2))) * u
.= (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((a2 * a3) * (c3 - c2)) * u) by RLVECT_1:def_7 ;
(((((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1) + (B1 * 1)) * o = ((((a3 * c3) - (a3 * c2)) * (B1 * (((a3 * c2) - (a2 * c2)) "))) + (B1 * ((((a3 * c2) - (a2 * c2)) ") * ((a3 * c2) - (a2 * c2))))) * o by A4, XCMPLX_0:def_7
.= ((B1 * (((a3 * c2) - (a2 * c2)) ")) * ((((a3 * c3) + (- (a3 * c2))) + (a3 * c2)) - (a2 * c2))) * o
.= (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((a3 * c3) - (a2 * c2)) * o) by RLVECT_1:def_7 ;
hence u1 = ((B1 * (((a3 * c2) - (a2 * c2)) ")) * ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u))) + ((B1 * (((a3 * c2) - (a2 * c2)) ")) * (((c2 * c3) * (a3 - a2)) * u2)) by A1, A3, A6, A5, RLVECT_1:def_5
.= (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2)) by RLVECT_1:def_5 ;
::_thesis: verum
end;

Lm27:  for V being RealLinearSpace
for p, q, r, u, u2, u1 being Element of V holds ((p + q) + r) + ((u + u2) + u1) = ((p + u) + (q + u2)) + (r + u1)
proof
let V be RealLinearSpace; ::_thesis: for p, q, r, u, u2, u1 being Element of V holds ((p + q) + r) + ((u + u2) + u1) = ((p + u) + (q + u2)) + (r + u1)
let p, q, r, u, u2, u1 be Element of V; ::_thesis: ((p + q) + r) + ((u + u2) + u1) = ((p + u) + (q + u2)) + (r + u1)
thus ((p + u) + (q + u2)) + (r + u1) = (u + (p + (q + u2))) + (r + u1) by RLVECT_1:def_3
.= (u + ((p + q) + u2)) + (r + u1) by RLVECT_1:def_3
.= ((u + u2) + (p + q)) + (r + u1) by RLVECT_1:def_3
.= (u + u2) + ((p + q) + (r + u1)) by RLVECT_1:def_3
.= (u + u2) + (((p + q) + r) + u1) by RLVECT_1:def_3
.= ((p + q) + r) + ((u + u2) + u1) by RLVECT_1:def_3 ; ::_thesis: verum
end;

Lm28:  for V being RealLinearSpace
for u1, o, u, u2, v1, w2 being Element of V
for a3, c3, a2, c2, C2, C3 being Real st u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) & v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) & C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) holds
((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V
proof
let V be RealLinearSpace; ::_thesis: for u1, o, u, u2, v1, w2 being Element of V
for a3, c3, a2, c2, C2, C3 being Real st u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) & v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) & C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) holds
((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V
let u1, o, u, u2, v1, w2 be Element of V; ::_thesis: for a3, c3, a2, c2, C2, C3 being Real st u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) & v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) & C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) holds
((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V
let a3, c3, a2, c2 be Real; ::_thesis: for C2, C3 being Real st u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) & v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) & C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) holds
((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V
let C2, C3 be Real; ::_thesis: ( u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) & v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) & C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) implies ((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V )
assume that
A1: u1 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2) and
A2: ( v1 = (o + (a3 * u)) + (c3 * u2) & w2 = (o + (a2 * u)) + (c2 * u2) ) and
A3: ( C2 + C3 = (a2 * c2) - (a3 * c3) & (C2 * a3) + (C3 * a2) = (a2 * a3) * (c2 - c3) & (C2 * c3) + (C3 * c2) = (c2 * c3) * (a2 - a3) ) ; ::_thesis: ((1 * u1) + (C2 * v1)) + (C3 * w2) = 0. V
A4: ((1 * u1) + (C2 * v1)) + (C3 * w2) = (u1 + (C2 * v1)) + (C3 * w2) by RLVECT_1:def_8
.= u1 + ((C2 * v1) + (C3 * w2)) by RLVECT_1:def_3 ;
(C2 * v1) + (C3 * w2) = ((C2 * (o + (a3 * u))) + (C2 * (c3 * u2))) + (C3 * ((o + (a2 * u)) + (c2 * u2))) by A2, RLVECT_1:def_5
.= (((C2 * o) + (C2 * (a3 * u))) + (C2 * (c3 * u2))) + (C3 * ((o + (a2 * u)) + (c2 * u2))) by RLVECT_1:def_5
.= (((C2 * o) + (C2 * (a3 * u))) + (C2 * (c3 * u2))) + ((C3 * (o + (a2 * u))) + (C3 * (c2 * u2))) by RLVECT_1:def_5
.= (((C2 * o) + (C2 * (a3 * u))) + (C2 * (c3 * u2))) + (((C3 * o) + (C3 * (a2 * u))) + (C3 * (c2 * u2))) by RLVECT_1:def_5
.= (((C2 * o) + (C3 * o)) + ((C2 * (a3 * u)) + (C3 * (a2 * u)))) + ((C2 * (c3 * u2)) + (C3 * (c2 * u2))) by Lm27
.= (((C2 + C3) * o) + ((C2 * (a3 * u)) + (C3 * (a2 * u)))) + ((C2 * (c3 * u2)) + (C3 * (c2 * u2))) by RLVECT_1:def_6
.= (((C2 + C3) * o) + (((C2 * a3) * u) + (C3 * (a2 * u)))) + ((C2 * (c3 * u2)) + (C3 * (c2 * u2))) by RLVECT_1:def_7
.= (((C2 + C3) * o) + (((C2 * a3) * u) + ((C3 * a2) * u))) + ((C2 * (c3 * u2)) + (C3 * (c2 * u2))) by RLVECT_1:def_7
.= (((C2 + C3) * o) + (((C2 * a3) * u) + ((C3 * a2) * u))) + (((C2 * c3) * u2) + (C3 * (c2 * u2))) by RLVECT_1:def_7
.= (((C2 + C3) * o) + (((C2 * a3) * u) + ((C3 * a2) * u))) + (((C2 * c3) * u2) + ((C3 * c2) * u2)) by RLVECT_1:def_7
.= (((C2 + C3) * o) + (((C2 * a3) + (C3 * a2)) * u)) + (((C2 * c3) * u2) + ((C3 * c2) * u2)) by RLVECT_1:def_6
.= ((((a2 * c2) - (a3 * c3)) * o) + (((a2 * a3) * (c2 - c3)) * u)) + (((c2 * c3) * (a2 - a3)) * u2) by A3, RLVECT_1:def_6 ;
hence ((1 * u1) + (C2 * v1)) + (C3 * w2) = (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * c2) - (a3 * c3)) * o)) + ((((a2 * a3) * (c3 - c2)) * u) + (((a2 * a3) * (c2 - c3)) * u))) + ((((c2 * c3) * (a3 - a2)) * u2) + (((c2 * c3) * (a2 - a3)) * u2)) by A1, A4, Lm27
.= (((((a3 * c3) - (a2 * c2)) + ((a2 * c2) - (a3 * c3))) * o) + ((((a2 * a3) * (c3 - c2)) * u) + (((a2 * a3) * (c2 - c3)) * u))) + ((((c2 * c3) * (a3 - a2)) * u2) + (((c2 * c3) * (a2 - a3)) * u2)) by RLVECT_1:def_6
.= (((((a3 * c3) - (a2 * c2)) + ((a2 * c2) - (a3 * c3))) * o) + ((((a2 * a3) * (c3 - c2)) + ((a2 * a3) * (c2 - c3))) * u)) + ((((c2 * c3) * (a3 - a2)) * u2) + (((c2 * c3) * (a2 - a3)) * u2)) by RLVECT_1:def_6
.= ((((((a3 * c3) + (- (a2 * c2))) + (a2 * c2)) + (- (a3 * c3))) * o) + ((((a2 * a3) * (c3 - c2)) + ((a2 * a3) * (c2 - c3))) * u)) + ((((c2 * c3) * (a3 - a2)) + ((c2 * c3) * (a2 - a3))) * u2) by RLVECT_1:def_6
.= ((0. V) + ((((a2 * a3) * (c3 - c2)) + ((a2 * a3) * (c2 - c3))) * u)) + ((((c2 * c3) * (a3 - a2)) + ((c2 * c3) * (a2 - a3))) * u2) by RLVECT_1:10
.= (0 * u) + ((((c2 * c3) * (a3 - a2)) + (- ((c2 * c3) * (a3 - a2)))) * u2) by RLVECT_1:4
.= (0. V) + (0 * u2) by RLVECT_1:10
.= (0. V) + (0. V) by RLVECT_1:10
.= 0. V by RLVECT_1:4 ;
::_thesis: verum
end;

Lm29:  for V being RealLinearSpace
for w2, o, u, u2, w1 being Element of V
for a2, c2, A3, A39, B3, B39 being Real st w2 = (o + (a2 * u)) + (c2 * u2) & w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 holds
w1 = B3 * w2
proof
let V be RealLinearSpace; ::_thesis: for w2, o, u, u2, w1 being Element of V
for a2, c2, A3, A39, B3, B39 being Real st w2 = (o + (a2 * u)) + (c2 * u2) & w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 holds
w1 = B3 * w2
let w2, o, u, u2, w1 be Element of V; ::_thesis: for a2, c2, A3, A39, B3, B39 being Real st w2 = (o + (a2 * u)) + (c2 * u2) & w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 holds
w1 = B3 * w2
let a2, c2 be Real; ::_thesis: for A3, A39, B3, B39 being Real st w2 = (o + (a2 * u)) + (c2 * u2) & w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 holds
w1 = B3 * w2
let A3, A39, B3, B39 be Real; ::_thesis: ( w2 = (o + (a2 * u)) + (c2 * u2) & w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 implies w1 = B3 * w2 )
assume that
A1: w2 = (o + (a2 * u)) + (c2 * u2) and
A2: ( w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) & B3 = B39 & A3 = B39 * a2 ) ; ::_thesis: w1 = B3 * w2
thus w1 = ((B3 * o) + (B3 * (a2 * u))) + ((B3 * c2) * u2) by A2, RLVECT_1:def_7
.= ((B3 * o) + (B3 * (a2 * u))) + (B3 * (c2 * u2)) by RLVECT_1:def_7
.= (B3 * (o + (a2 * u))) + (B3 * (c2 * u2)) by RLVECT_1:def_5
.= B3 * w2 by A1, RLVECT_1:def_5 ; ::_thesis: verum
end;

theorem Th9: :: ANPROJ_2:9
for V being RealLinearSpace
for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 lie_on_an_angle & o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop & u,v2,w1 are_LinDep & u2,v,w1 are_LinDep & u,w2,v1 are_LinDep & w,u2,v1 are_LinDep & v,w2,u1 are_LinDep & w,v2,u1 are_LinDep holds
u1,v1,w1 are_LinDep
proof
let V be RealLinearSpace; ::_thesis: for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 lie_on_an_angle & o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop & u,v2,w1 are_LinDep & u2,v,w1 are_LinDep & u,w2,v1 are_LinDep & w,u2,v1 are_LinDep & v,w2,u1 are_LinDep & w,v2,u1 are_LinDep holds
u1,v1,w1 are_LinDep
let o, u, v, w, u2, v2, w2, u1, v1, w1 be Element of V; ::_thesis: ( not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 lie_on_an_angle & o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop & u,v2,w1 are_LinDep & u2,v,w1 are_LinDep & u,w2,v1 are_LinDep & w,u2,v1 are_LinDep & v,w2,u1 are_LinDep & w,v2,u1 are_LinDep implies u1,v1,w1 are_LinDep )
assume that
A1: not o is zero and
A2: u,v,w are_Prop_Vect and
A3: u2,v2,w2 are_Prop_Vect and
A4: u1,v1,w1 are_Prop_Vect and
A5: o,u,v,w,u2,v2,w2 lie_on_an_angle and
A6: o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop and
A7: u,v2,w1 are_LinDep and
A8: u2,v,w1 are_LinDep and
A9: u,w2,v1 are_LinDep and
A10: w,u2,v1 are_LinDep and
A11: v,w2,u1 are_LinDep and
A12: w,v2,u1 are_LinDep ; ::_thesis: u1,v1,w1 are_LinDep
A13: not u is zero by A2, Def1;
A14: not are_Prop u2,v2 by A6, Def5;
A15: not are_Prop o,v by A6, Def5;
A16: not are_Prop u,v by A6, Def5;
A17: o,u2,v2 are_LinDep by A5, Def4;
A18: ( not are_Prop o,w2 & not are_Prop u2,w2 ) by A6, Def5;
A19: not u2 is zero by A3, Def1;
A20: ( not are_Prop o,w & not are_Prop u,w ) by A6, Def5;
A21: o,u,w are_LinDep by A5, Def4;
A22: not are_Prop o,v2 by A6, Def5;
A23: o,u2,w2 are_LinDep by A5, Def4;
A24: not o,u,u2 are_LinDep by A5, Def4;
then A25: not are_Prop o,u by ANPROJ_1:12;
A26: not w is zero by A2, Def1;
then  o,u,w are_Prop_Vect by A1, A13, Def1;
then consider a3, b3 being Real such that
A27: b3 * w = o + (a3 * u) and
 a3 <> 0 and
A28: b3 <> 0 by A21, A20, A25, Th6;
A29: not are_Prop u2,o by A24, ANPROJ_1:12;
A30: not w2 is zero by A3, Def1;
then  o,u2,w2 are_Prop_Vect by A1, A19, Def1;
then consider c3, d3 being Real such that
A31: d3 * w2 = o + (c3 * u2) and
 c3 <> 0 and
A32: d3 <> 0 by A23, A18, A29, Th6;
A33: o,u,v are_LinDep by A5, Def4;
A34: not v2 is zero by A3, Def1;
then  o,u2,v2 are_Prop_Vect by A1, A19, Def1;
then consider c2, d2 being Real such that
A35: d2 * v2 = o + (c2 * u2) and
A36: c2 <> 0 and
A37: d2 <> 0 by A17, A22, A14, A29, Th6;
A38: not v is zero by A2, Def1;
then  o,u,v are_Prop_Vect by A1, A13, Def1;
then consider a2, b2 being Real such that
A39: b2 * v = o + (a2 * u) and
 a2 <> 0 and
A40: b2 <> 0 by A33, A15, A16, A25, Th6;
set v9 = o + (a2 * u);
set w9 = o + (a3 * u);
set v29 = o + (c2 * u2);
set w29 = o + (c3 * u2);
A41: not o + (c2 * u2) is zero by A34, A35, A37, Lm20;
A42: not o + (a2 * u) is zero by A38, A39, A40, Lm20;
A43: not o + (a3 * u) is zero by A26, A27, A28, Lm20;
A44: not o + (c3 * u2) is zero by A30, A31, A32, Lm20;
A45: are_Prop w2,o + (c3 * u2) by A31, A32, Lm16;
then A46: u,o + (c3 * u2),v1 are_LinDep by A9, ANPROJ_1:4;
A47: are_Prop v,o + (a2 * u) by A39, A40, Lm16;
then A48: o + (a2 * u),o + (c3 * u2),u1 are_LinDep by A11, A45, ANPROJ_1:4;
A49: are_Prop v2,o + (c2 * u2) by A35, A37, Lm16;
then A50: u,o + (c2 * u2),w1 are_LinDep by A7, ANPROJ_1:4;
A51: are_Prop w,o + (a3 * u) by A27, A28, Lm16;
then A52: o + (a3 * u),o + (c2 * u2),u1 are_LinDep by A12, A49, ANPROJ_1:4;
 not are_Prop u,v2
proof
assume  are_Prop u,v2 ; ::_thesis: contradiction
then  o,u2,u are_LinDep by A17, ANPROJ_1:4;
hence  contradiction by A24, ANPROJ_1:5; ::_thesis: verum
end;
then  not are_Prop u,o + (c2 * u2) by A35, A37, Lm17;
then consider A3, B3 being Real such that
A53: w1 = (A3 * u) + (B3 * (o + (c2 * u2))) by A13, A50, A41, ANPROJ_1:6;
 not o,u2,v are_LinDep
proof
assume  o,u2,v are_LinDep ; ::_thesis: contradiction
then A54: o,v,u2 are_LinDep by ANPROJ_1:5;
 ( o,v,u are_LinDep & o,v,o are_LinDep ) by A33, ANPROJ_1:5, ANPROJ_1:11;
hence  contradiction by A1, A38, A24, A15, A54, ANPROJ_1:14; ::_thesis: verum
end;
then  not are_Prop v,w2 by A23, ANPROJ_1:4;
then  not are_Prop o + (a2 * u),o + (c3 * u2) by A39, A40, A31, A32, Lm21;
then consider A1, B1 being Real such that
A55: u1 = (A1 * (o + (a2 * u))) + (B1 * (o + (c3 * u2))) by A42, A44, A48, ANPROJ_1:6;
 not o,u,v2 are_LinDep
proof
assume  o,u,v2 are_LinDep ; ::_thesis: contradiction
then A56: o,v2,u are_LinDep by ANPROJ_1:5;
 ( o,v2,u2 are_LinDep & o,v2,o are_LinDep ) by A17, ANPROJ_1:5, ANPROJ_1:11;
hence  contradiction by A1, A34, A24, A22, A56, ANPROJ_1:14; ::_thesis: verum
end;
then  not are_Prop v2,w by A21, ANPROJ_1:4;
then  not are_Prop o + (a3 * u),o + (c2 * u2) by A27, A28, A35, A37, Lm21;
then consider A19, B19 being Real such that
A57: u1 = (A19 * (o + (a3 * u))) + (B19 * (o + (c2 * u2))) by A41, A43, A52, ANPROJ_1:6;
A58: u1 = (((A1 + B1) * o) + ((A1 * a2) * u)) + ((B1 * c3) * u2) by A55, Lm22;
A59: not are_Prop v2,w2 by A6, Def5;
A60: not are_Prop v,w by A6, Def5;
A61: ( not are_Prop o + (a2 * u),o + (a3 * u) & not are_Prop o + (c2 * u2),o + (c3 * u2) )
proof
A62: now__::_thesis:_not_are_Prop_o_+_(c2_*_u2),o_+_(c3_*_u2)
assume  are_Prop o + (c2 * u2),o + (c3 * u2) ; ::_thesis: contradiction
then  are_Prop v2,o + (c3 * u2) by A49, ANPROJ_1:2;
hence  contradiction by A59, A45, ANPROJ_1:2; ::_thesis: verum
end;
A63: now__::_thesis:_not_are_Prop_o_+_(a2_*_u),o_+_(a3_*_u)
assume  are_Prop o + (a2 * u),o + (a3 * u) ; ::_thesis: contradiction
then  are_Prop v,o + (a3 * u) by A47, ANPROJ_1:2;
hence  contradiction by A60, A51, ANPROJ_1:2; ::_thesis: verum
end;
assume  ( are_Prop o + (a2 * u),o + (a3 * u) or are_Prop o + (c2 * u2),o + (c3 * u2) ) ; ::_thesis: contradiction
hence  contradiction by A63, A62; ::_thesis: verum
end;
 not are_Prop u,w2
proof
assume  are_Prop u,w2 ; ::_thesis: contradiction
then  o,u2,u are_LinDep by A23, ANPROJ_1:4;
hence  contradiction by A24, ANPROJ_1:5; ::_thesis: verum
end;
then  not are_Prop u,o + (c3 * u2) by A31, A32, Lm17;
then consider A2, B2 being Real such that
A64: v1 = (A2 * u) + (B2 * (o + (c3 * u2))) by A13, A44, A46, ANPROJ_1:6;
 u2,w,v1 are_LinDep by A10, ANPROJ_1:5;
then A65: u2,o + (a3 * u),v1 are_LinDep by A51, ANPROJ_1:4;
 not are_Prop u2,w by A24, A21, ANPROJ_1:4;
then  not are_Prop u2,o + (a3 * u) by A27, A28, Lm17;
then consider A29, B29 being Real such that
A66: v1 = (A29 * u2) + (B29 * (o + (a3 * u))) by A19, A43, A65, ANPROJ_1:6;
A67: v1 = ((B2 * o) + (A2 * u)) + ((B2 * c3) * u2) by A64, Lm18;
A68: u2,o + (a2 * u),w1 are_LinDep by A8, A47, ANPROJ_1:4;
 not are_Prop u2,v by A24, A33, ANPROJ_1:4;
then  not are_Prop u2,o + (a2 * u) by A39, A40, Lm17;
then consider A39, B39 being Real such that
A69: w1 = (A39 * u2) + (B39 * (o + (a2 * u))) by A19, A68, A42, ANPROJ_1:6;
A70: w1 = ((B3 * o) + (A3 * u)) + ((B3 * c2) * u2) by A53, Lm18;
 v1 = ((B29 * o) + ((B29 * a3) * u)) + (A29 * u2) by A66, Lm19;
then A71: ( B2 = B29 & A2 = B29 * a3 ) by A24, A67, ANPROJ_1:8;
 w1 = ((B39 * o) + ((B39 * a2) * u)) + (A39 * u2) by A69, Lm19;
then A72: ( B3 = B39 & A3 = B39 * a2 ) by A24, A70, ANPROJ_1:8;
A73: u1 = (((A19 + B19) * o) + ((A19 * a3) * u)) + ((B19 * c2) * u2) by A57, Lm22;
then A74: B1 * c3 = B19 * c2 by A24, A58, ANPROJ_1:8;
 ( A1 + B1 = A19 + B19 & A1 * a2 = A19 * a3 ) by A24, A58, A73, ANPROJ_1:8;
then A75: A1 = (((a3 * c3) - (a3 * c2)) * (((a3 * c2) - (a2 * c2)) ")) * B1 by A36, A61, A74, Lm24;
set v19 = (o + (a3 * u)) + (c3 * u2);
set C2 = a2 * c2;
set C3 = - (a3 * c3);
set u19 = ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2);
set w19 = (o + (a2 * u)) + (c2 * u2);
A76: ((a2 * c2) * c3) + ((- (a3 * c3)) * c2) = (c2 * c3) * (a2 - a3) ;
 ( (a2 * c2) + (- (a3 * c3)) = (a2 * c2) - (a3 * c3) & ((a2 * c2) * a3) + ((- (a3 * c3)) * a2) = (a2 * a3) * (c2 - c3) ) ;
then  ((1 * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2))) + ((a2 * c2) * ((o + (a3 * u)) + (c3 * u2)))) + ((- (a3 * c3)) * ((o + (a2 * u)) + (c2 * u2))) = 0. V by A76, Lm28;
then A77: ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2),(o + (a3 * u)) + (c3 * u2),(o + (a2 * u)) + (c2 * u2) are_LinDep by ANPROJ_1:def_2;
A78: not v1 is zero by A4, Def1;
A79: B2 <> 0
proof
assume  not B2 <> 0 ; ::_thesis: contradiction
then v1 = (0. V) + (0 * (o + (c3 * u2))) by A64, A71, RLVECT_1:10
.= (0. V) + (0. V) by RLVECT_1:10
.= 0. V by RLVECT_1:4 ;
hence  contradiction by A78; ::_thesis: verum
end;
 v1 = B2 * ((o + (a3 * u)) + (c3 * u2)) by A67, A71, Lm29;
then A80: are_Prop (o + (a3 * u)) + (c3 * u2),v1 by A79, Lm16;
A81: not w1 is zero by A4, Def1;
A82: B3 <> 0
proof
assume  not B3 <> 0 ; ::_thesis: contradiction
then w1 = (0. V) + (0 * (o + (c2 * u2))) by A53, A72, RLVECT_1:10
.= (0. V) + (0. V) by RLVECT_1:10
.= 0. V by RLVECT_1:4 ;
hence  contradiction by A81; ::_thesis: verum
end;
 w1 = B3 * ((o + (a2 * u)) + (c2 * u2)) by A70, A72, Lm29;
then A83: are_Prop (o + (a2 * u)) + (c2 * u2),w1 by A82, Lm16;
A84: not u1 is zero by A4, Def1;
A85: B1 <> 0
proof
assume  not B1 <> 0 ; ::_thesis: contradiction
then u1 = (0. V) + (0 * (o + (c3 * u2))) by A55, A75, RLVECT_1:10
.= (0. V) + (0. V) by RLVECT_1:10
.= 0. V by RLVECT_1:4 ;
hence  contradiction by A84; ::_thesis: verum
end;
 u1 = (B1 * (((a3 * c2) - (a2 * c2)) ")) * (((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2)) by A36, A61, A58, A75, Lm26;
then  are_Prop ((((a3 * c3) - (a2 * c2)) * o) + (((a2 * a3) * (c3 - c2)) * u)) + (((c2 * c3) * (a3 - a2)) * u2),u1 by A36, A61, A85, Lm16, Lm25;
hence  u1,v1,w1 are_LinDep by A83, A80, A77, ANPROJ_1:4; ::_thesis: verum
end;


theorem Th10: :: ANPROJ_2:10
for A being non empty set
for x1 being Element of A ex f being Element of Funcs (A,REAL) st
( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) )
proof
let A be non empty set ; ::_thesis: for x1 being Element of A ex f being Element of Funcs (A,REAL) st
( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) )
let x1 be Element of A; ::_thesis: ex f being Element of Funcs (A,REAL) st
( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) )
deffunc H1( set ) ->  Element of NAT = 0 ;
deffunc H2( set ) ->  Element of NAT = 1;
defpred S1[ set ] means $1 = x1;
A1: for z being set st z in A holds
( ( S1[z] implies H2(z) in REAL ) & ( not S1[z] implies H1(z) in REAL ) ) ;
consider f being Function of A,REAL such that
A2: for z being set st z in A holds
( ( S1[z] implies f . z = H2(z) ) & ( not S1[z] implies f . z = H1(z) ) ) from FUNCT_2:sch_5(A1);
reconsider f = f as Element of Funcs (A,REAL) by FUNCT_2:8;
take  f ; ::_thesis: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) )
thus  ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by A2; ::_thesis: verum
end;

theorem Th11: :: ANPROJ_2:11
for A being non empty set
for f, g, h being Element of Funcs (A,REAL)
for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
proof
let A be non empty set ; ::_thesis: for f, g, h being Element of Funcs (A,REAL)
for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let f, g, h be Element of Funcs (A,REAL); ::_thesis: for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let x1, x2, x3 be Element of A; ::_thesis: ( x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) implies for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) )
set RM = RealFuncExtMult A;
set RA = RealFuncAdd A;
assume that
A1: x1 <> x2 and
A2: x1 <> x3 and
A3: x2 <> x3 and
A4: f . x1 = 1 and
A5: for z being set st z in A & z <> x1 holds
f . z = 0 and
A6: g . x2 = 1 and
A7: for z being set st z in A & z <> x2 holds
g . z = 0 and
A8: h . x3 = 1 and
A9: for z being set st z in A & z <> x3 holds
h . z = 0 ; ::_thesis: for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
A10: ( f . x2 = 0 & h . x2 = 0 ) by A1, A3, A5, A9;
let a, b, c be Real; ::_thesis: ( (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 ) )
assume A11: (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A ; ::_thesis: ( a = 0 & b = 0 & c = 0 )
then A12: 0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x2 by FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x2) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2)) by FUNCSDOM:4
.= ((a * 0) + (b * 1)) + (c * 0) by A6, A10, FUNCSDOM:4
.= b ;
A13: ( g . x1 = 0 & h . x1 = 0 ) by A1, A2, A7, A9;
A14: ( f . x3 = 0 & g . x3 = 0 ) by A2, A3, A5, A7;
A15: 0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x3 by A11, FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x3) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3)) by FUNCSDOM:4
.= ((a * 0) + (b * 0)) + (c * 1) by A8, A14, FUNCSDOM:4
.= c ;
0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x1 by A11, FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x1) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1)) by FUNCSDOM:4
.= ((a * 1) + (b * 0)) + (c * 0) by A4, A13, FUNCSDOM:4
.= a ;
hence  ( a = 0 & b = 0 & c = 0 ) by A12, A15; ::_thesis: verum
end;

theorem :: ANPROJ_2:12
for A being non empty set
for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
proof
let A be non empty set ; ::_thesis: for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let x1, x2, x3 be Element of A; ::_thesis: ( x1 <> x2 & x1 <> x3 & x2 <> x3 implies ex f, g, h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) )
assume A1: ( x1 <> x2 & x1 <> x3 & x2 <> x3 ) ; ::_thesis: ex f, g, h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A3: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A4: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
take  g ; ::_thesis: ex h being Element of Funcs (A,REAL) st
for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
take  h ; ::_thesis: for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let a, b, c be Real; ::_thesis: ( (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 ) )
assume  (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A ; ::_thesis: ( a = 0 & b = 0 & c = 0 )
hence  ( a = 0 & b = 0 & c = 0 ) by A1, A2, A4, A3, Th11; ::_thesis: verum
end;

theorem Th13: :: ANPROJ_2:13
for A being non empty set
for f, g, h being Element of Funcs (A,REAL)
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
proof
let A be non empty set ; ::_thesis: for f, g, h being Element of Funcs (A,REAL)
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
let f, g, h be Element of Funcs (A,REAL); ::_thesis: for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
let x1, x2, x3 be Element of A; ::_thesis: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) implies for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) )
assume that
A1: A = {x1,x2,x3} and
A2: x1 <> x2 and
A3: x1 <> x3 and
A4: x2 <> x3 and
A5: f . x1 = 1 and
A6: for z being set st z in A & z <> x1 holds
f . z = 0 and
A7: g . x2 = 1 and
A8: for z being set st z in A & z <> x2 holds
g . z = 0 and
A9: h . x3 = 1 and
A10: for z being set st z in A & z <> x3 holds
h . z = 0 ; ::_thesis: for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
A11: ( g . x1 = 0 & h . x1 = 0 ) by A2, A3, A8, A10;
A12: ( f . x2 = 0 & h . x2 = 0 ) by A2, A4, A6, A10;
let h9 be Element of Funcs (A,REAL); ::_thesis: ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
take a = h9 . x1; ::_thesis: ex b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
take b = h9 . x2; ::_thesis: ex c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
take c = h9 . x3; ::_thesis: h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
A13: ( f . x3 = 0 & g . x3 = 0 ) by A3, A4, A6, A8;
now__::_thesis:_for_x_being_Element_of_A_holds_h9_._x_=_((RealFuncAdd_A)_._(((RealFuncAdd_A)_._(((RealFuncExtMult_A)_._[a,f]),((RealFuncExtMult_A)_._[b,g]))),((RealFuncExtMult_A)_._[c,h])))_._x
let x be Element of A; ::_thesis: h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x
A14: ( x = x1 or x = x2 or x = x3 ) by A1, ENUMSET1:def_1;
A15: ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x2 = (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x2) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2)) by FUNCSDOM:4
.= ((a * 0) + (b * 1)) + (c * 0) by A7, A12, FUNCSDOM:4
.= h9 . x2 ;
A16: ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x3 = (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x3) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3)) by FUNCSDOM:4
.= ((a * 0) + (b * 0)) + (c * 1) by A9, A13, FUNCSDOM:4
.= h9 . x3 ;
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x1 = (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x1) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:1
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1)) by FUNCSDOM:4
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1)) by FUNCSDOM:4
.= ((a * 1) + (b * 0)) + (c * 0) by A5, A11, FUNCSDOM:4
.= h9 . x1 ;
hence  h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x by A14, A15, A16; ::_thesis: verum
end;
hence  h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) by FUNCT_2:63; ::_thesis: verum
end;

theorem :: ANPROJ_2:14
for A being non empty set
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
proof
let A be non empty set ; ::_thesis: for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
let x1, x2, x3 be Element of A; ::_thesis: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 implies ex f, g, h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) )
assume A1: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 ) ; ::_thesis: ex f, g, h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A3: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A4: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
take  g ; ::_thesis: ex h being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
take  h ; ::_thesis: for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
let h9 be Element of Funcs (A,REAL); ::_thesis: ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
thus  ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) by A1, A2, A4, A3, Th13; ::_thesis: verum
end;

theorem Th15: :: ANPROJ_2:15
for A being non empty set
for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
proof
let A be non empty set ; ::_thesis: for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds
ex f, g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
let x1, x2, x3 be Element of A; ::_thesis: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 implies ex f, g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) ) )
assume A1: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 ) ; ::_thesis: ex f, g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A3: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A4: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
take  g ; ::_thesis: ex h being Element of Funcs (A,REAL) st
( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
take  h ; ::_thesis: ( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
thus  ( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) ) by A1, A2, A4, A3, Th11, Th13; ::_thesis: verum
end;

Lm30:  ex A being non empty set ex x1, x2, x3 being Element of A st
( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 )
proof
reconsider A = {0,1,2} as non empty set ;
take  A ; ::_thesis: ex x1, x2, x3 being Element of A st
( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 )
reconsider x1 = 0 , x2 = 1, x3 = 2 as Element of A by ENUMSET1:def_1;
take  x1 ; ::_thesis: ex x2, x3 being Element of A st
( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 )
take  x2 ; ::_thesis: ex x3 being Element of A st
( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 )
take  x3 ; ::_thesis: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 )
thus  ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 ) ; ::_thesis: verum
end;

theorem Th16: :: ANPROJ_2:16
ex V being non trivial RealLinearSpace ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
proof
consider A being non empty set , x1, x2, x3 being Element of A such that
A1: ( A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 ) by Lm30;
set V = RealVectSpace A;
consider f, g, h being Element of Funcs (A,REAL) such that
A2: for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) and
A3: for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) by A1, Th15;
reconsider u = f, v = g, w = h as Element of (RealVectSpace A) ;
 for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. (RealVectSpace A) holds
( a = 0 & b = 0 & c = 0 ) by A2;
then  not u is zero by Th1;
then A4: u <> 0. (RealVectSpace A) ;
A5: for y being Element of (RealVectSpace A) ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w)
proof
let y be Element of (RealVectSpace A); ::_thesis: ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w)
reconsider h9 = y as Element of Funcs (A,REAL) ;
consider a, b, c being Real such that
A6: h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) by A3;
 h9 = ((a * u) + (b * v)) + (c * w) by A6;
hence  ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ; ::_thesis: verum
end;
reconsider V = RealVectSpace A as non trivial RealLinearSpace by A4, STRUCT_0:def_18;
take  V ; ::_thesis: ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
reconsider u = u, v = v, w = w as Element of V ;
take  u ; ::_thesis: ex v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
take  v ; ::_thesis: ex w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
take  w ; ::_thesis: ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
thus  ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) by A2, A5; ::_thesis: verum
end;

theorem Th17: :: ANPROJ_2:17
for A being non empty set
for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
proof
let A be non empty set ; ::_thesis: for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
let f, g, h, f1 be Element of Funcs (A,REAL); ::_thesis: for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
let x1, x2, x3, x4 be Element of A; ::_thesis: ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) implies for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) )
set RM = RealFuncExtMult A;
set RA = RealFuncAdd A;
assume that
A1: x1 <> x2 and
A2: x1 <> x3 and
A3: x1 <> x4 and
A4: x2 <> x3 and
A5: x2 <> x4 and
A6: x3 <> x4 and
A7: f . x1 = 1 and
A8: for z being set st z in A & z <> x1 holds
f . z = 0 and
A9: g . x2 = 1 and
A10: for z being set st z in A & z <> x2 holds
g . z = 0 and
A11: h . x3 = 1 and
A12: for z being set st z in A & z <> x3 holds
h . z = 0 and
A13: f1 . x4 = 1 and
A14: for z being set st z in A & z <> x4 holds
f1 . z = 0 ; ::_thesis: for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
A15: ( f . x2 = 0 & h . x2 = 0 ) by A1, A4, A8, A12;
A16: ( g . x1 = 0 & h . x1 = 0 ) by A1, A2, A10, A12;
A17: f1 . x1 = 0 by A3, A14;
A18: f1 . x2 = 0 by A5, A14;
let a, b, c, d be Real; ::_thesis: ( (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 & d = 0 ) )
assume A19: (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A ; ::_thesis: ( a = 0 & b = 0 & c = 0 & d = 0 )
then A20: 0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x2 by FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x2) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x2) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((a * 0) + (b * 1)) + (c * 0)) + (d * 0) by A9, A15, A18, FUNCSDOM:4
.= b ;
A21: ( f . x4 = 0 & g . x4 = 0 ) by A3, A5, A8, A10;
A22: h . x4 = 0 by A6, A12;
A23: ( f . x3 = 0 & g . x3 = 0 ) by A2, A4, A8, A10;
A24: f1 . x3 = 0 by A6, A14;
A25: 0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x4 by A19, FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x4) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x4) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (b * (g . x4))) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 0)) + (d * 1) by A13, A21, A22, FUNCSDOM:4
.= d ;
A26: 0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x3 by A19, FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x3) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x3) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 1)) + (d * 0) by A11, A23, A24, FUNCSDOM:4
.= c ;
0 = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x1 by A19, FUNCOP_1:7
.= (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x1) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x1) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((a * 1) + (b * 0)) + (c * 0)) + (d * 0) by A7, A16, A17, FUNCSDOM:4
.= a ;
hence  ( a = 0 & b = 0 & c = 0 & d = 0 ) by A20, A26, A25; ::_thesis: verum
end;

theorem :: ANPROJ_2:18
for A being non empty set
for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
proof
let A be non empty set ; ::_thesis: for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
let x1, x2, x3, x4 be Element of A; ::_thesis: ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 implies ex f, g, h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) )
assume A1: ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) ; ::_thesis: ex f, g, h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider f1 being Element of Funcs (A,REAL) such that
A3: ( f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A4: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A5: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
take  g ; ::_thesis: ex h, f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
take  h ; ::_thesis: ex f1 being Element of Funcs (A,REAL) st
for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
take  f1 ; ::_thesis: for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )
let a, b, c, d be Real; ::_thesis: ( (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 & d = 0 ) )
assume  (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A ; ::_thesis: ( a = 0 & b = 0 & c = 0 & d = 0 )
hence  ( a = 0 & b = 0 & c = 0 & d = 0 ) by A1, A2, A5, A4, A3, Th17; ::_thesis: verum
end;

theorem Th19: :: ANPROJ_2:19
for A being non empty set
for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
proof
let A be non empty set ; ::_thesis: for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let f, g, h, f1 be Element of Funcs (A,REAL); ::_thesis: for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let x1, x2, x3, x4 be Element of A; ::_thesis: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) implies for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) )
assume that
A1: A = {x1,x2,x3,x4} and
A2: x1 <> x2 and
A3: x1 <> x3 and
A4: x1 <> x4 and
A5: x2 <> x3 and
A6: x2 <> x4 and
A7: x3 <> x4 and
A8: f . x1 = 1 and
A9: for z being set st z in A & z <> x1 holds
f . z = 0 and
A10: g . x2 = 1 and
A11: for z being set st z in A & z <> x2 holds
g . z = 0 and
A12: h . x3 = 1 and
A13: for z being set st z in A & z <> x3 holds
h . z = 0 and
A14: f1 . x4 = 1 and
A15: for z being set st z in A & z <> x4 holds
f1 . z = 0 ; ::_thesis: for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
A16: ( f . x4 = 0 & g . x4 = 0 ) by A4, A6, A9, A11;
A17: f1 . x3 = 0 by A7, A15;
A18: f1 . x2 = 0 by A6, A15;
A19: ( f . x2 = 0 & h . x2 = 0 ) by A2, A5, A9, A13;
A20: f1 . x1 = 0 by A4, A15;
A21: ( g . x1 = 0 & h . x1 = 0 ) by A2, A3, A11, A13;
A22: h . x4 = 0 by A7, A13;
let h9 be Element of Funcs (A,REAL); ::_thesis: ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take a = h9 . x1; ::_thesis: ex b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take b = h9 . x2; ::_thesis: ex c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take c = h9 . x3; ::_thesis: ex d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take d = h9 . x4; ::_thesis: h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
A23: ( f . x3 = 0 & g . x3 = 0 ) by A3, A5, A9, A11;
now__::_thesis:_for_x_being_Element_of_A_holds_h9_._x_=_((RealFuncAdd_A)_._(((RealFuncAdd_A)_._(((RealFuncAdd_A)_._(((RealFuncExtMult_A)_._[a,f]),((RealFuncExtMult_A)_._[b,g]))),((RealFuncExtMult_A)_._[c,h]))),((RealFuncExtMult_A)_._[d,f1])))_._x
let x be Element of A; ::_thesis: h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x
A24: ( x = x1 or x = x2 or x = x3 or x = x4 ) by A1, ENUMSET1:def_2;
A25: ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x2 = (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x2) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x2) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((a * 0) + (b * 1)) + (c * 0)) + (d * 0) by A10, A19, A18, FUNCSDOM:4
.= h9 . x2 ;
A26: ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x4 = (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x4) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x4) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x4) + (b * (g . x4))) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 0)) + (d * 1) by A14, A16, A22, FUNCSDOM:4
.= h9 . x4 ;
A27: ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x3 = (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x3) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x3) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 1)) + (d * 0) by A12, A23, A17, FUNCSDOM:4
.= h9 . x3 ;
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x1 = (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x1) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= ((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x1) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1) by FUNCSDOM:1
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((a * 1) + (b * 0)) + (c * 0)) + (d * 0) by A8, A21, A20, FUNCSDOM:4
.= h9 . x1 ;
hence  h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x by A24, A25, A27, A26; ::_thesis: verum
end;
hence  h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) by FUNCT_2:63; ::_thesis: verum
end;

theorem :: ANPROJ_2:20
for A being non empty set
for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
proof
let A be non empty set ; ::_thesis: for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let x1, x2, x3, x4 be Element of A; ::_thesis: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 implies ex f, g, h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) )
assume A1: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) ; ::_thesis: ex f, g, h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider f1 being Element of Funcs (A,REAL) such that
A3: ( f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A4: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A5: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take  g ; ::_thesis: ex h, f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take  h ; ::_thesis: ex f1 being Element of Funcs (A,REAL) st
for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take  f1 ; ::_thesis: for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let h9 be Element of Funcs (A,REAL); ::_thesis: ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
thus  ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) by A1, A2, A5, A4, A3, Th19; ::_thesis: verum
end;

theorem Th21: :: ANPROJ_2:21
for A being non empty set
for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
proof
let A be non empty set ; ::_thesis: for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds
ex f, g, h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
let x1, x2, x3, x4 be Element of A; ::_thesis: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 implies ex f, g, h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) ) )
assume A1: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) ; ::_thesis: ex f, g, h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
consider f being Element of Funcs (A,REAL) such that
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) by Th10;
consider f1 being Element of Funcs (A,REAL) such that
A3: ( f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) ) by Th10;
consider h being Element of Funcs (A,REAL) such that
A4: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) by Th10;
consider g being Element of Funcs (A,REAL) such that
A5: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) by Th10;
take  f ; ::_thesis: ex g, h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
take  g ; ::_thesis: ex h, f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
take  h ; ::_thesis: ex f1 being Element of Funcs (A,REAL) st
( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
take  f1 ; ::_thesis: ( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
thus  ( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) ) by A1, A2, A5, A4, A3, Th17, Th19; ::_thesis: verum
end;

Lm31:  ex A being non empty set ex x1, x2, x3, x4 being Element of A st
( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
proof
reconsider A = {0,1,2,3} as non empty set ;
take  A ; ::_thesis: ex x1, x2, x3, x4 being Element of A st
( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
reconsider x1 = 0 , x2 = 1, x3 = 2, x4 = 3 as Element of A by ENUMSET1:def_2;
take  x1 ; ::_thesis: ex x2, x3, x4 being Element of A st
( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
take  x2 ; ::_thesis: ex x3, x4 being Element of A st
( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
take  x3 ; ::_thesis: ex x4 being Element of A st
( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
take  x4 ; ::_thesis: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
thus  ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) ; ::_thesis: verum
end;

theorem Th22: :: ANPROJ_2:22
ex V being non trivial RealLinearSpace ex u, v, w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
proof
consider A being non empty set , x1, x2, x3, x4 being Element of A such that
A1: ( A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) by Lm31;
set V = RealVectSpace A;
consider f, g, h, f1 being Element of Funcs (A,REAL) such that
A2: for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) and
A3: for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) by A1, Th21;
reconsider u = f, v = g, w = h, u1 = f1 as Element of (RealVectSpace A) ;
 for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. (RealVectSpace A) holds
( a = 0 & b = 0 & c = 0 & d = 0 ) by A2;
then  not u is zero by Th2;
then A4: u <> 0. (RealVectSpace A) ;
A5: for y being Element of (RealVectSpace A) ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1)
proof
let y be Element of (RealVectSpace A); ::_thesis: ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1)
reconsider h9 = y as Element of Funcs (A,REAL) ;
consider a, b, c, d being Real such that
A6: h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) by A3;
 h9 = (((a * u) + (b * v)) + (c * w)) + (d * u1) by A6;
hence  ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ; ::_thesis: verum
end;
reconsider V = RealVectSpace A as non trivial RealLinearSpace by A4, STRUCT_0:def_18;
take  V ; ::_thesis: ex u, v, w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
reconsider u = u, v = v, w = w, u1 = u1 as Element of V ;
take  u ; ::_thesis: ex v, w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
take  v ; ::_thesis: ex w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
take  w ; ::_thesis: ex u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
take  u1 ; ::_thesis: ( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
thus  ( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) ) by A2, A5; ::_thesis: verum
end;

definition
let IT be RealLinearSpace;
attrIT is up-3-dimensional  means :Def6: :: ANPROJ_2:def 6
 ex u, v, w1 being Element of IT st
for a, b, c being Real st ((a * u) + (b * v)) + (c * w1) = 0. IT holds
( a = 0 & b = 0 & c = 0 );
end;

:: deftheorem Def6 defines up-3-dimensional ANPROJ_2:def_6_:_
 for IT being RealLinearSpace holds
( IT is up-3-dimensional iff ex u, v, w1 being Element of IT st
for a, b, c being Real st ((a * u) + (b * v)) + (c * w1) = 0. IT holds
( a = 0 & b = 0 & c = 0 ) );

registration
cluster non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional for RLSStruct ;
existence
 ex b1 being RealLinearSpace st b1 is up-3-dimensional
proof
consider V0 being non trivial RealLinearSpace such that
A1: ex u, v, w being Element of V0 st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V0 holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V0 ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) by Th16;
take  V0 ; ::_thesis: V0 is up-3-dimensional
thus  V0 is up-3-dimensional by A1, Def6; ::_thesis: verum
end;
end;

registration
cluster non empty V70() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital up-3-dimensional  ->  non trivial for RLSStruct ;
coherence
 for b1 being RealLinearSpace st b1 is up-3-dimensional holds
not b1 is trivial
proof
let V be RealLinearSpace; ::_thesis: ( V is up-3-dimensional implies not V is trivial )
given u, v, w1 being Element of V such that A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w1) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; :: according to ANPROJ_2:def_6 ::_thesis: not V is trivial
now__::_thesis:_not_w1_=_0._V
assume  w1 = 0. V ; ::_thesis: contradiction
then A2: 0. V = 1 * w1 by RLVECT_1:10;
 ( 0. V = 0 * u & 0. V = 0 * v ) by RLVECT_1:10;
then  0. V = (0 * u) + (0 * v) by RLVECT_1:4
.= ((0 * u) + (0 * v)) + (1 * w1) by A2, RLVECT_1:4 ;
hence  contradiction by A1; ::_thesis: verum
end;
hence  not V is trivial by STRUCT_0:def_18; ::_thesis: verum
end;
end;

definition
let CS be non empty CollStr ;
redefine attr CS is reflexive  means :Def7: :: ANPROJ_2:def 7
 for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear );
compatibility
 ( CS is reflexive iff for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) )
proof
hereby  ::_thesis: ( ( for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) ) implies CS is reflexive )
assume A1: CS is reflexive ; ::_thesis: for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear )
let p, q, r be Element of CS; ::_thesis: ( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear )
 [p,q,p] in the Collinearity of CS by A1, COLLSP:def_3;
hence  p,q,p is_collinear by COLLSP:def_2; ::_thesis: ( p,p,q is_collinear & p,q,q is_collinear )
 [p,p,q] in the Collinearity of CS by A1, COLLSP:def_3;
hence  p,p,q is_collinear by COLLSP:def_2; ::_thesis: p,q,q is_collinear
 [p,q,q] in the Collinearity of CS by A1, COLLSP:def_3;
hence  p,q,q is_collinear by COLLSP:def_2; ::_thesis: verum
end;
assume A2: for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) ; ::_thesis: CS is reflexive
let p, q, r be Element of CS; :: according to COLLSP:def_3 ::_thesis: ( ( not p = q & not p = r & not q = r ) or [p,q,r] in the Collinearity of CS )
assume A3: ( p = q or p = r or q = r ) ; ::_thesis: [p,q,r] in the Collinearity of CS
percases ( p = q or p = r or q = r ) by A3;
suppose p = q ; ::_thesis: [p,q,r] in the Collinearity of CS
then  p,q,r is_collinear by A2;
hence  [p,q,r] in the Collinearity of CS by COLLSP:def_2; ::_thesis: verum
end;
suppose p = r ; ::_thesis: [p,q,r] in the Collinearity of CS
then  p,q,r is_collinear by A2;
hence  [p,q,r] in the Collinearity of CS by COLLSP:def_2; ::_thesis: verum
end;
suppose q = r ; ::_thesis: [p,q,r] in the Collinearity of CS
then  p,q,r is_collinear by A2;
hence  [p,q,r] in the Collinearity of CS by COLLSP:def_2; ::_thesis: verum
end;
end;
end;
redefine attr CS is transitive  means :Def8: :: ANPROJ_2:def 8
 for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear ;
compatibility
 ( CS is transitive iff for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear )
proof
hereby  ::_thesis: ( ( for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear ) implies CS is transitive )
assume A4: CS is transitive ; ::_thesis: for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear
let p, q, r, r1, r2 be Element of CS; ::_thesis: ( p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear implies r,r1,r2 is_collinear )
assume that
A5: p <> q and
A6: ( p,q,r is_collinear & p,q,r1 is_collinear ) and
A7: p,q,r2 is_collinear ; ::_thesis: r,r1,r2 is_collinear
A8: [p,q,r2] in the Collinearity of CS by A7, COLLSP:def_2;
 ( [p,q,r] in the Collinearity of CS & [p,q,r1] in the Collinearity of CS ) by A6, COLLSP:def_2;
then  [r,r1,r2] in the Collinearity of CS by A4, A5, A8, COLLSP:def_4;
hence  r,r1,r2 is_collinear by COLLSP:def_2; ::_thesis: verum
end;
assume A9: for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear ; ::_thesis: CS is transitive
let p, q, r, r1, r2 be Element of CS; :: according to COLLSP:def_4 ::_thesis: ( p = q or not [p,q,r] in the Collinearity of CS or not [p,q,r1] in the Collinearity of CS or not [p,q,r2] in the Collinearity of CS or [r,r1,r2] in the Collinearity of CS )
assume that
A10: p <> q and
A11: ( [p,q,r] in the Collinearity of CS & [p,q,r1] in the Collinearity of CS ) and
A12: [p,q,r2] in the Collinearity of CS ; ::_thesis: [r,r1,r2] in the Collinearity of CS
A13: p,q,r2 is_collinear by A12, COLLSP:def_2;
 ( p,q,r is_collinear & p,q,r1 is_collinear ) by A11, COLLSP:def_2;
then  r,r1,r2 is_collinear by A9, A10, A13;
hence  [r,r1,r2] in the Collinearity of CS by COLLSP:def_2; ::_thesis: verum
end;
end;

:: deftheorem Def7 defines reflexive ANPROJ_2:def_7_:_
 for CS being non empty CollStr holds
( CS is reflexive iff for p, q, r being Element of CS holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) );

:: deftheorem Def8 defines transitive ANPROJ_2:def_8_:_
 for CS being non empty CollStr holds
( CS is transitive iff for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear );

definition
let IT be non empty CollStr ;
attrIT is Vebleian  means :Def9: :: ANPROJ_2:def 9
 for p, p1, p2, r, r1 being Element of IT st p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of IT st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear );
attrIT is at_least_3rank  means :Def10: :: ANPROJ_2:def 10
 for p, q being Element of IT ex r being Element of IT st
( p <> r & q <> r & p,q,r is_collinear );
end;

:: deftheorem Def9 defines Vebleian ANPROJ_2:def_9_:_
 for IT being non empty CollStr holds
( IT is Vebleian iff for p, p1, p2, r, r1 being Element of IT st p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of IT st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) );

:: deftheorem Def10 defines at_least_3rank ANPROJ_2:def_10_:_
 for IT being non empty CollStr holds
( IT is at_least_3rank iff for p, q being Element of IT ex r being Element of IT st
( p <> r & q <> r & p,q,r is_collinear ) );


theorem Th23: :: ANPROJ_2:23
for V being non trivial RealLinearSpace
for p, q, r being Element of (ProjectiveSpace V) holds
( p,q,r is_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) )
proof
let V be non trivial RealLinearSpace; ::_thesis: for p, q, r being Element of (ProjectiveSpace V) holds
( p,q,r is_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) )
let p, q, r be Element of (ProjectiveSpace V); ::_thesis: ( p,q,r is_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) )
A1: now__::_thesis:_(_p,q,r_is_collinear_implies_ex_u,_v,_w_being_Element_of_V_st_
(_p_=_Dir_u_&_q_=_Dir_v_&_r_=_Dir_w_&_not_u_is_zero_&_not_v_is_zero_&_not_w_is_zero_&_u,v,w_are_LinDep_)_)
assume  p,q,r is_collinear ; ::_thesis: ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep )
then  [p,q,r] in the Collinearity of (ProjectiveSpace V) by COLLSP:def_2;
then consider u, v, w being Element of V such that
A2: ( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) by ANPROJ_1:24;
take u = u; ::_thesis: ex v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep )
take v = v; ::_thesis: ex w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep )
take w = w; ::_thesis: ( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep )
thus  ( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) by A2; ::_thesis: verum
end;
now__::_thesis:_(_ex_u,_v,_w_being_Element_of_V_st_
(_p_=_Dir_u_&_q_=_Dir_v_&_r_=_Dir_w_&_not_u_is_zero_&_not_v_is_zero_&_not_w_is_zero_&_u,v,w_are_LinDep_)_implies_p,q,r_is_collinear_)
assume  ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) ; ::_thesis: p,q,r is_collinear
then  [p,q,r] in the Collinearity of (ProjectiveSpace V) by ANPROJ_1:25;
hence  p,q,r is_collinear by COLLSP:def_2; ::_thesis: verum
end;
hence  ( p,q,r is_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) ) by A1; ::_thesis: verum
end;

Lm32:  for V being non trivial RealLinearSpace holds ProjectiveSpace V is reflexive
proof
let V be non trivial RealLinearSpace; ::_thesis: ProjectiveSpace V is reflexive
let p be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_7 ::_thesis: for q, r being Element of (ProjectiveSpace V) holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear )
let q be Element of (ProjectiveSpace V); ::_thesis: for r being Element of (ProjectiveSpace V) holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear )
consider u being Element of V such that
A1: ( not u is zero & p = Dir u ) by ANPROJ_1:26;
consider v being Element of V such that
A2: ( not v is zero & q = Dir v ) by ANPROJ_1:26;
A3: u,v,v are_LinDep by ANPROJ_1:11;
 ( u,v,u are_LinDep & u,u,v are_LinDep ) by ANPROJ_1:11;
hence  for r being Element of (ProjectiveSpace V) holds
( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) by A1, A2, A3, Th23; ::_thesis: verum
end;

Lm33:  for V being non trivial RealLinearSpace holds ProjectiveSpace V is transitive
proof
let V be non trivial RealLinearSpace; ::_thesis: ProjectiveSpace V is transitive
let p be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_8 ::_thesis: for q, r, r1, r2 being Element of (ProjectiveSpace V) st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds
r,r1,r2 is_collinear
let q, r, r1, r2 be Element of (ProjectiveSpace V); ::_thesis: ( p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear implies r,r1,r2 is_collinear )
assume that
A1: p <> q and
A2: p,q,r is_collinear and
A3: p,q,r1 is_collinear and
A4: p,q,r2 is_collinear ; ::_thesis: r,r1,r2 is_collinear
consider u1, v1, w1 being Element of V such that
A5: p = Dir u1 and
A6: q = Dir v1 and
A7: r = Dir w1 and
A8: not u1 is zero and
A9: not v1 is zero and
A10: not w1 is zero and
A11: u1,v1,w1 are_LinDep by A2, Th23;
consider v being Element of V such that
A12: not v is zero and
A13: q = Dir v by ANPROJ_1:26;
A14: are_Prop v1,v by A12, A13, A6, A9, ANPROJ_1:22;
consider u3, v3, w3 being Element of V such that
A15: p = Dir u3 and
A16: q = Dir v3 and
A17: r2 = Dir w3 and
A18: not u3 is zero and
A19: not v3 is zero and
A20: not w3 is zero and
A21: u3,v3,w3 are_LinDep by A4, Th23;
A22: are_Prop v3,v by A12, A13, A16, A19, ANPROJ_1:22;
consider u2, v2, w2 being Element of V such that
A23: p = Dir u2 and
A24: q = Dir v2 and
A25: r1 = Dir w2 and
A26: not u2 is zero and
A27: not v2 is zero and
A28: not w2 is zero and
A29: u2,v2,w2 are_LinDep by A3, Th23;
A30: are_Prop v2,v by A12, A13, A24, A27, ANPROJ_1:22;
consider u being Element of V such that
A31: not u is zero and
A32: p = Dir u by ANPROJ_1:26;
 are_Prop u1,u by A31, A32, A5, A8, ANPROJ_1:22;
then A33: u,v,w1 are_LinDep by A11, A14, ANPROJ_1:4;
 are_Prop u3,u by A31, A32, A15, A18, ANPROJ_1:22;
then A34: u,v,w3 are_LinDep by A21, A22, ANPROJ_1:4;
 are_Prop u2,u by A31, A32, A23, A26, ANPROJ_1:22;
then A35: u,v,w2 are_LinDep by A29, A30, ANPROJ_1:4;
 not are_Prop u,v by A1, A31, A32, A12, A13, ANPROJ_1:22;
then  w1,w2,w3 are_LinDep by A31, A12, A33, A35, A34, ANPROJ_1:14;
hence  r,r1,r2 is_collinear by A7, A10, A25, A28, A17, A20, Th23; ::_thesis: verum
end;

registration
let V be non trivial RealLinearSpace;
cluster ProjectiveSpace V ->  reflexive transitive ;
coherence
 ( ProjectiveSpace V is reflexive & ProjectiveSpace V is transitive ) by Lm32, Lm33;
end;

theorem Th24: :: ANPROJ_2:24
for V being non trivial RealLinearSpace
for p, q, r being Element of (ProjectiveSpace V) st p,q,r is_collinear holds
( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear )
proof
let V be non trivial RealLinearSpace; ::_thesis: for p, q, r being Element of (ProjectiveSpace V) st p,q,r is_collinear holds
( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear )
let p, q, r be Element of (ProjectiveSpace V); ::_thesis: ( p,q,r is_collinear implies ( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear ) )
assume  p,q,r is_collinear ; ::_thesis: ( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear )
then consider u, v, w being Element of V such that
A1: ( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero ) and
A2: u,v,w are_LinDep by Th23;
A3: ( w,v,u are_LinDep & w,u,v are_LinDep ) by A2, ANPROJ_1:5;
A4: v,w,u are_LinDep by A2, ANPROJ_1:5;
 ( u,w,v are_LinDep & v,u,w are_LinDep ) by A2, ANPROJ_1:5;
hence  ( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear ) by A1, A3, A4, Th23; ::_thesis: verum
end;

Lm34:  for V being non trivial RealLinearSpace
for p, p1, p2, r, r1 being Element of (ProjectiveSpace V) st p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
proof
let V be non trivial RealLinearSpace; ::_thesis: for p, p1, p2, r, r1 being Element of (ProjectiveSpace V) st p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
let p, p1, p2, r, r1 be Element of (ProjectiveSpace V); ::_thesis: ( p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear implies ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) )
assume that
A1: p,p1,p2 is_collinear and
A2: p,p1,r is_collinear and
A3: p1,p2,r1 is_collinear ; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
A4: now__::_thesis:_(_p_<>_p2_implies_ex_r2_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p2,r2_is_collinear_&_r,r1,r2_is_collinear_)_)
A5: now__::_thesis:_(_p1_<>_p_implies_ex_r,_r2_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p2,r2_is_collinear_&_r,r1,r2_is_collinear_)_)
assume A6: p1 <> p ; ::_thesis: ex r, r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
take r = r; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
A7: r,r1,r is_collinear by Def7;
 p,p1,p is_collinear by Def7;
then  p,p2,r is_collinear by A1, A2, A6, Def8;
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A7; ::_thesis: verum
end;
A8: now__::_thesis:_(_p1_<>_p2_implies_ex_r1,_r2_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p2,r2_is_collinear_&_r,r1,r2_is_collinear_)_)
assume A9: p1 <> p2 ; ::_thesis: ex r1, r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
take r1 = r1; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
A10: r,r1,r1 is_collinear by Def7;
 ( p1,p2,p is_collinear & p1,p2,p2 is_collinear ) by A1, Def7, Th24;
then  p,p2,r1 is_collinear by A3, A9, Def8;
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A10; ::_thesis: verum
end;
assume  p <> p2 ; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A5, A8; ::_thesis: verum
end;
now__::_thesis:_(_p_=_p2_implies_ex_r,_r2_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p2,r2_is_collinear_&_r,r1,r2_is_collinear_)_)
assume  p = p2 ; ::_thesis: ex r, r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
then A11: p,p2,r is_collinear by Def7;
take r = r; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
 r,r1,r is_collinear by Def7;
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A11; ::_thesis: verum
end;
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A4; ::_thesis: verum
end;

Lm35:  for V being non trivial RealLinearSpace
for p, p1, p2, r, r1 being Element of (ProjectiveSpace V) st not p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
proof
let V be non trivial RealLinearSpace; ::_thesis: for p, p1, p2, r, r1 being Element of (ProjectiveSpace V) st not p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
let p, p1, p2, r, r1 be Element of (ProjectiveSpace V); ::_thesis: ( not p,p1,p2 is_collinear & p,p1,r is_collinear & p1,p2,r1 is_collinear implies ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) )
assume that
A1: not p,p1,p2 is_collinear and
A2: p,p1,r is_collinear and
A3: p1,p2,r1 is_collinear ; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
consider u, v, t being Element of V such that
A4: p = Dir u and
A5: p1 = Dir v and
A6: r = Dir t and
A7: not u is zero and
A8: not v is zero and
A9: not t is zero and
A10: u,v,t are_LinDep by A2, Th23;
consider v1, w, s being Element of V such that
A11: p1 = Dir v1 and
A12: p2 = Dir w and
A13: r1 = Dir s and
A14: not v1 is zero and
A15: not w is zero and
A16: not s is zero and
A17: v1,w,s are_LinDep by A3, Th23;
 are_Prop v1,v by A5, A8, A11, A14, ANPROJ_1:22;
then A18: v,w,s are_LinDep by A17, ANPROJ_1:4;
 not u,v,w are_LinDep by A1, A4, A5, A7, A8, A12, A15, Th23;
then consider y being Element of V such that
A19: ( u,w,y are_LinDep & t,s,y are_LinDep ) and
A20: not y is zero by A10, A18, ANPROJ_1:15;
reconsider r2 = Dir y as Element of (ProjectiveSpace V) by A20, ANPROJ_1:26;
take  r2 ; ::_thesis: ( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
thus  ( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A4, A6, A7, A9, A12, A13, A15, A16, A19, A20, Th23; ::_thesis: verum
end;

Lm36:  for V being non trivial RealLinearSpace holds ProjectiveSpace V is Vebleian
proof
let V be non trivial RealLinearSpace; ::_thesis: ProjectiveSpace V is Vebleian
let p be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_9 ::_thesis: for p1, p2, r, r1 being Element of (ProjectiveSpace V) st p,p1,r is_collinear & p1,p2,r1 is_collinear holds
ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
let p1, p2, r, r1 be Element of (ProjectiveSpace V); ::_thesis: ( p,p1,r is_collinear & p1,p2,r1 is_collinear implies ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) )
assume A1: ( p,p1,r is_collinear & p1,p2,r1 is_collinear ) ; ::_thesis: ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
then  ( p,p1,p2 is_collinear implies ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) ) by Lm34;
hence  ex r2 being Element of (ProjectiveSpace V) st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A1, Lm35; ::_thesis: verum
end;

registration
let V be non trivial RealLinearSpace;
cluster ProjectiveSpace V ->  Vebleian ;
coherence
 ProjectiveSpace V is Vebleian by Lm36;
end;

Lm37:  for V being non trivial RealLinearSpace st V is up-3-dimensional holds
ProjectiveSpace V is proper
proof
let V be non trivial RealLinearSpace; ::_thesis: ( V is up-3-dimensional implies ProjectiveSpace V is proper )
given u, v, w being Element of V such that A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; :: according to ANPROJ_2:def_6 ::_thesis: ProjectiveSpace V is proper
A2: not u,v,w are_LinDep by A1, Th1;
A3: not w is zero by A1, Th1;
A4: ( not u is zero & not v is zero ) by A1, Th1;
then reconsider p = Dir u, q = Dir v, r = Dir w as Element of (ProjectiveSpace V) by A3, ANPROJ_1:26;
take  p ; :: according to COLLSP:def_6 ::_thesis: not for b1, b2 being Element of the U1 of (ProjectiveSpace V) holds p,b1,b2 is_collinear
take  q ; ::_thesis: not for b1 being Element of the U1 of (ProjectiveSpace V) holds p,q,b1 is_collinear
take  r ; ::_thesis: not p,q,r is_collinear
assume  p,q,r is_collinear ; ::_thesis: contradiction
then  [(Dir u),(Dir v),(Dir w)] in the Collinearity of (ProjectiveSpace V) by COLLSP:def_2;
hence  contradiction by A4, A3, A2, ANPROJ_1:25; ::_thesis: verum
end;

registration
let V be up-3-dimensional RealLinearSpace;
cluster ProjectiveSpace V ->  proper ;
coherence
 ProjectiveSpace V is proper by Lm37;
end;

theorem Th25: :: ANPROJ_2:25
for V being non trivial RealLinearSpace st ex u, v being Element of V st
for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) holds
ProjectiveSpace V is at_least_3rank
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex u, v being Element of V st
for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) implies ProjectiveSpace V is at_least_3rank )
given u, v being Element of V such that A1: for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ; ::_thesis: ProjectiveSpace V is at_least_3rank
A2: not are_Prop u,v by A1, Lm1;
let p be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_10 ::_thesis: for q being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p <> r & q <> r & p,q,r is_collinear )
let q be Element of (ProjectiveSpace V); ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p <> r & q <> r & p,q,r is_collinear )
consider y being Element of V such that
A3: ( not y is zero & p = Dir y ) by ANPROJ_1:26;
consider w being Element of V such that
A4: ( not w is zero & q = Dir w ) by ANPROJ_1:26;
 ( not u is zero & not v is zero ) by A1, Lm1;
then consider z being Element of V such that
A5: not z is zero and
A6: y,w,z are_LinDep and
A7: not are_Prop y,z and
A8: not are_Prop w,z by A2, ANPROJ_1:16;
reconsider r = Dir z as Element of (ProjectiveSpace V) by A5, ANPROJ_1:26;
take  r ; ::_thesis: ( p <> r & q <> r & p,q,r is_collinear )
thus  p <> r by A3, A5, A7, ANPROJ_1:22; ::_thesis: ( q <> r & p,q,r is_collinear )
thus  q <> r by A4, A5, A8, ANPROJ_1:22; ::_thesis: p,q,r is_collinear
thus  p,q,r is_collinear by A3, A4, A5, A6, Th23; ::_thesis: verum
end;

Lm38:  for V being non trivial RealLinearSpace st V is up-3-dimensional holds
ProjectiveSpace V is at_least_3rank
proof
let V be non trivial RealLinearSpace; ::_thesis: ( V is up-3-dimensional implies ProjectiveSpace V is at_least_3rank )
given u, v, w1 being Element of V such that A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w1) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; :: according to ANPROJ_2:def_6 ::_thesis: ProjectiveSpace V is at_least_3rank
now__::_thesis:_for_a,_b_being_Real_st_(a_*_u)_+_(b_*_v)_=_0._V_holds_
(_a_=_0_&_b_=_0_)
let a, b be Real; ::_thesis: ( (a * u) + (b * v) = 0. V implies ( a = 0 & b = 0 ) )
assume  (a * u) + (b * v) = 0. V ; ::_thesis: ( a = 0 & b = 0 )
then  0. V = ((a * u) + (b * v)) + (0. V) by RLVECT_1:4
.= ((a * u) + (b * v)) + (0 * w1) by RLVECT_1:10 ;
hence  ( a = 0 & b = 0 ) by A1; ::_thesis: verum
end;
hence  ProjectiveSpace V is at_least_3rank by Th25; ::_thesis: verum
end;

registration
let V be up-3-dimensional RealLinearSpace;
cluster ProjectiveSpace V ->  at_least_3rank ;
coherence
 ProjectiveSpace V is at_least_3rank by Lm38;
end;

registration
cluster non empty reflexive transitive proper Vebleian at_least_3rank for CollStr ;
existence
 ex b1 being non empty CollStr st
( b1 is transitive & b1 is reflexive & b1 is proper & b1 is Vebleian & b1 is at_least_3rank )
proof
set V0 = the up-3-dimensional RealLinearSpace;
take  ProjectiveSpace the up-3-dimensional RealLinearSpace ; ::_thesis: ( ProjectiveSpace the up-3-dimensional RealLinearSpace is transitive & ProjectiveSpace the up-3-dimensional RealLinearSpace is reflexive & ProjectiveSpace the up-3-dimensional RealLinearSpace is proper & ProjectiveSpace the up-3-dimensional RealLinearSpace is Vebleian & ProjectiveSpace the up-3-dimensional RealLinearSpace is at_least_3rank )
thus  ( ProjectiveSpace the up-3-dimensional RealLinearSpace is transitive & ProjectiveSpace the up-3-dimensional RealLinearSpace is reflexive & ProjectiveSpace the up-3-dimensional RealLinearSpace is proper & ProjectiveSpace the up-3-dimensional RealLinearSpace is Vebleian & ProjectiveSpace the up-3-dimensional RealLinearSpace is at_least_3rank ) ; ::_thesis: verum
end;
end;

definition
mode CollProjectiveSpace is non empty reflexive transitive proper Vebleian at_least_3rank CollStr ;
end;

definition
let IT be CollProjectiveSpace;
attrIT is Fanoian  means :Def11: :: ANPROJ_2:def 11
 for p1, r2, q, r1, q1, p, r being Element of IT st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds
r2,r1,q1 is_collinear ;
attrIT is Desarguesian  means :: ANPROJ_2:def 12
 for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> q1 & p1 <> q1 & o <> q2 & p2 <> q2 & o <> q3 & p3 <> q3 & not o,p1,p2 is_collinear & not o,p1,p3 is_collinear & not o,p2,p3 is_collinear & p1,p2,r3 is_collinear & q1,q2,r3 is_collinear & p2,p3,r1 is_collinear & q2,q3,r1 is_collinear & p1,p3,r2 is_collinear & q1,q3,r2 is_collinear & o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear holds
r1,r2,r3 is_collinear ;
attrIT is Pappian  means :: ANPROJ_2:def 13
 for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> p2 & o <> p3 & p2 <> p3 & p1 <> p2 & p1 <> p3 & o <> q2 & o <> q3 & q2 <> q3 & q1 <> q2 & q1 <> q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear & o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear & p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear & p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear holds
r1,r2,r3 is_collinear ;
end;

:: deftheorem Def11 defines Fanoian ANPROJ_2:def_11_:_
 for IT being CollProjectiveSpace holds
( IT is Fanoian iff for p1, r2, q, r1, q1, p, r being Element of IT st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds
r2,r1,q1 is_collinear );

:: deftheorem  defines Desarguesian ANPROJ_2:def_12_:_
 for IT being CollProjectiveSpace holds
( IT is Desarguesian iff for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> q1 & p1 <> q1 & o <> q2 & p2 <> q2 & o <> q3 & p3 <> q3 & not o,p1,p2 is_collinear & not o,p1,p3 is_collinear & not o,p2,p3 is_collinear & p1,p2,r3 is_collinear & q1,q2,r3 is_collinear & p2,p3,r1 is_collinear & q2,q3,r1 is_collinear & p1,p3,r2 is_collinear & q1,q3,r2 is_collinear & o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear holds
r1,r2,r3 is_collinear );

:: deftheorem  defines Pappian ANPROJ_2:def_13_:_
 for IT being CollProjectiveSpace holds
( IT is Pappian iff for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> p2 & o <> p3 & p2 <> p3 & p1 <> p2 & p1 <> p3 & o <> q2 & o <> q3 & q2 <> q3 & q1 <> q2 & q1 <> q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear & o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear & p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear & p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear holds
r1,r2,r3 is_collinear );

definition
let IT be CollProjectiveSpace;
attrIT is 2-dimensional  means :Def14: :: ANPROJ_2:def 14
 for p, p1, q, q1 being Element of IT ex r being Element of IT st
( p,p1,r is_collinear & q,q1,r is_collinear );
end;

:: deftheorem Def14 defines 2-dimensional ANPROJ_2:def_14_:_
 for IT being CollProjectiveSpace holds
( IT is 2-dimensional iff for p, p1, q, q1 being Element of IT ex r being Element of IT st
( p,p1,r is_collinear & q,q1,r is_collinear ) );

notation
let IT be CollProjectiveSpace;
antonym  up-3-dimensional IT for  2-dimensional ;
end;

definition
let IT be CollProjectiveSpace;
attrIT is at_most-3-dimensional  means :Def15: :: ANPROJ_2:def 15
 for p, p1, q, q1, r2 being Element of IT ex r, r1 being Element of IT st
( p,q,r is_collinear & p1,q1,r1 is_collinear & r2,r,r1 is_collinear );
end;

:: deftheorem Def15 defines at_most-3-dimensional ANPROJ_2:def_15_:_
 for IT being CollProjectiveSpace holds
( IT is at_most-3-dimensional iff for p, p1, q, q1, r2 being Element of IT ex r, r1 being Element of IT st
( p,q,r is_collinear & p1,q1,r1 is_collinear & r2,r,r1 is_collinear ) );

theorem Th26: :: ANPROJ_2:26
for V being non trivial RealLinearSpace
for p1, r2, q, r1, q1, p, r being Element of (ProjectiveSpace V) st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds
r2,r1,q1 is_collinear
proof
let V be non trivial RealLinearSpace; ::_thesis: for p1, r2, q, r1, q1, p, r being Element of (ProjectiveSpace V) st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds
r2,r1,q1 is_collinear
let p1, r2, q, r1, q1, p, r be Element of (ProjectiveSpace V); ::_thesis: ( p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear implies r2,r1,q1 is_collinear )
assume that
A1: p1,r2,q is_collinear and
A2: r1,q1,q is_collinear and
A3: p1,r1,p is_collinear and
A4: r2,q1,p is_collinear and
A5: p1,q1,r is_collinear and
A6: r2,r1,r is_collinear and
A7: p,q,r is_collinear ; ::_thesis: ( p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or p1,r1,q1 is_collinear or r2,r1,q1 is_collinear )
consider u1, w1, x being Element of V such that
A8: p1 = Dir u1 and
A9: r1 = Dir w1 and
A10: p = Dir x and
A11: not u1 is zero and
A12: not w1 is zero and
A13: not x is zero and
A14: u1,w1,x are_LinDep by A3, Th23;
consider u, v, z being Element of V such that
A15: p1 = Dir u and
A16: r2 = Dir v and
A17: q = Dir z and
A18: not u is zero and
A19: not v is zero and
A20: not z is zero and
A21: u,v,z are_LinDep by A1, Th23;
consider w, y, z1 being Element of V such that
A22: r1 = Dir w and
A23: q1 = Dir y and
A24: q = Dir z1 and
A25: not w is zero and
A26: not y is zero and
A27: not z1 is zero and
A28: w,y,z1 are_LinDep by A2, Th23;
A29: are_Prop w1,w by A22, A25, A9, A12, ANPROJ_1:22;
 are_Prop z1,z by A17, A20, A24, A27, ANPROJ_1:22;
then A30: w,y,z are_LinDep by A28, ANPROJ_1:4;
consider x2, z2, t2 being Element of V such that
A31: p = Dir x2 and
A32: q = Dir z2 and
A33: r = Dir t2 and
A34: not x2 is zero and
A35: not z2 is zero and
A36: not t2 is zero and
A37: x2,z2,t2 are_LinDep by A7, Th23;
A38: are_Prop x2,x by A10, A13, A31, A34, ANPROJ_1:22;
consider u2, y2, t being Element of V such that
A39: p1 = Dir u2 and
A40: q1 = Dir y2 and
A41: r = Dir t and
A42: not u2 is zero and
A43: not y2 is zero and
A44: not t is zero and
A45: u2,y2,t are_LinDep by A5, Th23;
A46: are_Prop y2,y by A23, A26, A40, A43, ANPROJ_1:22;
A47: are_Prop t2,t by A41, A44, A33, A36, ANPROJ_1:22;
 are_Prop z2,z by A17, A20, A32, A35, ANPROJ_1:22;
then A48: x,z,t are_LinDep by A37, A38, A47, ANPROJ_1:4;
 are_Prop u2,u by A15, A18, A39, A42, ANPROJ_1:22;
then A49: u,y,t are_LinDep by A45, A46, ANPROJ_1:4;
consider v2, w2, t1 being Element of V such that
A50: r2 = Dir v2 and
A51: r1 = Dir w2 and
A52: r = Dir t1 and
A53: not v2 is zero and
A54: not w2 is zero and
A55: not t1 is zero and
A56: v2,w2,t1 are_LinDep by A6, Th23;
A57: are_Prop t1,t by A41, A44, A52, A55, ANPROJ_1:22;
 are_Prop u1,u by A15, A18, A8, A11, ANPROJ_1:22;
then A58: u,w,x are_LinDep by A14, A29, ANPROJ_1:4;
consider v1, y1, x1 being Element of V such that
A59: r2 = Dir v1 and
A60: q1 = Dir y1 and
A61: p = Dir x1 and
A62: not v1 is zero and
A63: not y1 is zero and
A64: not x1 is zero and
A65: v1,y1,x1 are_LinDep by A4, Th23;
A66: are_Prop x1,x by A10, A13, A61, A64, ANPROJ_1:22;
A67: are_Prop w2,w by A22, A25, A51, A54, ANPROJ_1:22;
 are_Prop v2,v by A16, A19, A50, A53, ANPROJ_1:22;
then A68: v,w,t are_LinDep by A56, A67, A57, ANPROJ_1:4;
A69: are_Prop y1,y by A23, A26, A60, A63, ANPROJ_1:22;
 are_Prop v1,v by A16, A19, A59, A62, ANPROJ_1:22;
then  v,y,x are_LinDep by A65, A69, A66, ANPROJ_1:4;
then  ( u,v,y are_LinDep or u,v,w are_LinDep or u,w,y are_LinDep or v,w,y are_LinDep ) by A20, A21, A13, A44, A30, A58, A49, A68, A48, ANPROJ_1:18;
hence  ( p1,r2,q1 is_collinear or p1,r2,r1 is_collinear or p1,r1,q1 is_collinear or r2,r1,q1 is_collinear ) by A15, A16, A18, A19, A22, A23, A25, A26, Th23; ::_thesis: verum
end;

Lm39:  for V being up-3-dimensional RealLinearSpace holds ProjectiveSpace V is Fanoian
proof
let V be up-3-dimensional RealLinearSpace; ::_thesis: ProjectiveSpace V is Fanoian
 for p1, r2, q, r1, q1, p, r being Element of (ProjectiveSpace V) st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds
r2,r1,q1 is_collinear by Th26;
hence  ProjectiveSpace V is Fanoian by Def11; ::_thesis: verum
end;

Lm40:  for V being up-3-dimensional RealLinearSpace holds ProjectiveSpace V is Desarguesian
proof
let V be up-3-dimensional RealLinearSpace; ::_thesis: ProjectiveSpace V is Desarguesian
let o, p1, p2, p3, q1, q2, q3, r1, r2, r3 be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_12 ::_thesis: ( o <> q1 & p1 <> q1 & o <> q2 & p2 <> q2 & o <> q3 & p3 <> q3 & not o,p1,p2 is_collinear & not o,p1,p3 is_collinear & not o,p2,p3 is_collinear & p1,p2,r3 is_collinear & q1,q2,r3 is_collinear & p2,p3,r1 is_collinear & q2,q3,r1 is_collinear & p1,p3,r2 is_collinear & q1,q3,r2 is_collinear & o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear implies r1,r2,r3 is_collinear )
assume that
A1: o <> q1 and
A2: p1 <> q1 and
A3: o <> q2 and
A4: p2 <> q2 and
A5: o <> q3 and
A6: p3 <> q3 and
A7: not o,p1,p2 is_collinear and
A8: ( not o,p1,p3 is_collinear & not o,p2,p3 is_collinear ) and
A9: p1,p2,r3 is_collinear and
A10: q1,q2,r3 is_collinear and
A11: p2,p3,r1 is_collinear and
A12: q2,q3,r1 is_collinear and
A13: p1,p3,r2 is_collinear and
A14: q1,q3,r2 is_collinear and
A15: o,p1,q1 is_collinear and
A16: o,p2,q2 is_collinear and
A17: o,p3,q3 is_collinear ; ::_thesis: r1,r2,r3 is_collinear
consider q19, q29, r399 being Element of V such that
A18: q1 = Dir q19 and
A19: q2 = Dir q29 and
A20: r3 = Dir r399 and
A21: not q19 is zero and
A22: not q29 is zero and
A23: not r399 is zero and
A24: q19,q29,r399 are_LinDep by A10, Th23;
consider q299, q39, r199 being Element of V such that
A25: q2 = Dir q299 and
A26: q3 = Dir q39 and
A27: r1 = Dir r199 and
A28: not q299 is zero and
A29: not q39 is zero and
A30: not r199 is zero and
A31: q299,q39,r199 are_LinDep by A12, Th23;
A32: are_Prop q299,q29 by A19, A22, A25, A28, ANPROJ_1:22;
consider p299, p39, r19 being Element of V such that
A33: p2 = Dir p299 and
A34: p3 = Dir p39 and
A35: r1 = Dir r19 and
A36: not p299 is zero and
A37: not p39 is zero and
A38: not r19 is zero and
A39: p299,p39,r19 are_LinDep by A11, Th23;
A40: not are_Prop p39,q39 by A6, A34, A37, A26, A29, ANPROJ_1:22;
 are_Prop r199,r19 by A35, A38, A27, A30, ANPROJ_1:22;
then A41: q29,q39,r19 are_LinDep by A31, A32, ANPROJ_1:4;
consider p199, p399, r29 being Element of V such that
A42: p1 = Dir p199 and
A43: p3 = Dir p399 and
A44: r2 = Dir r29 and
A45: not p199 is zero and
A46: not p399 is zero and
A47: not r29 is zero and
A48: p199,p399,r29 are_LinDep by A13, Th23;
A49: are_Prop p399,p39 by A34, A37, A43, A46, ANPROJ_1:22;
consider o9 being Element of V such that
A50: not o9 is zero and
A51: o = Dir o9 by ANPROJ_1:26;
A52: not are_Prop o9,q39 by A5, A50, A51, A26, A29, ANPROJ_1:22;
consider p19, p29, r39 being Element of V such that
A53: p1 = Dir p19 and
A54: p2 = Dir p29 and
A55: r3 = Dir r39 and
A56: not p19 is zero and
A57: not p29 is zero and
A58: not r39 is zero and
A59: p19,p29,r39 are_LinDep by A9, Th23;
A60: ( not are_Prop p19,q19 & not are_Prop p29,q29 ) by A2, A4, A53, A54, A56, A57, A18, A19, A21, A22, ANPROJ_1:22;
A61: ( not are_Prop o9,q19 & not are_Prop o9,q29 ) by A1, A3, A50, A51, A18, A19, A21, A22, ANPROJ_1:22;
consider o999, p2999, q2999 being Element of V such that
A62: o = Dir o999 and
A63: p2 = Dir p2999 and
A64: q2 = Dir q2999 and
A65: not o999 is zero and
A66: not p2999 is zero and
A67: not q2999 is zero and
A68: o999,p2999,q2999 are_LinDep by A16, Th23;
A69: are_Prop q2999,q29 by A19, A22, A64, A67, ANPROJ_1:22;
A70: are_Prop o999,o9 by A50, A51, A62, A65, ANPROJ_1:22;
 are_Prop p2999,p29 by A54, A57, A63, A66, ANPROJ_1:22;
then A71: o9,p29,q29 are_LinDep by A68, A70, A69, ANPROJ_1:4;
consider q199, q399, r299 being Element of V such that
A72: q1 = Dir q199 and
A73: q3 = Dir q399 and
A74: r2 = Dir r299 and
A75: not q199 is zero and
A76: not q399 is zero and
A77: not r299 is zero and
A78: q199,q399,r299 are_LinDep by A14, Th23;
A79: are_Prop q199,q19 by A18, A21, A72, A75, ANPROJ_1:22;
A80: not o9,p19,p29 are_LinDep by A7, A50, A51, A53, A54, A56, A57, Th23;
 are_Prop r399,r39 by A55, A58, A20, A23, ANPROJ_1:22;
then A81: q19,q29,r39 are_LinDep by A24, ANPROJ_1:4;
A82: are_Prop q399,q39 by A26, A29, A73, A76, ANPROJ_1:22;
consider o9999, p3999, q3999 being Element of V such that
A83: o = Dir o9999 and
A84: p3 = Dir p3999 and
A85: q3 = Dir q3999 and
A86: not o9999 is zero and
A87: not p3999 is zero and
A88: not q3999 is zero and
A89: o9999,p3999,q3999 are_LinDep by A17, Th23;
A90: are_Prop q3999,q39 by A26, A29, A85, A88, ANPROJ_1:22;
 are_Prop p299,p29 by A54, A57, A33, A36, ANPROJ_1:22;
then A91: p29,p39,r19 are_LinDep by A39, ANPROJ_1:4;
 are_Prop p199,p19 by A53, A56, A42, A45, ANPROJ_1:22;
then  p19,p39,r29 are_LinDep by A48, A49, ANPROJ_1:4;
then A92: p19,p29,p39,r19,r29,r39 lie_on_a_triangle by A59, A91, Def2;
A93: q19,q29,q39 are_Prop_Vect by A21, A22, A29, Def1;
consider o99, p1999, q1999 being Element of V such that
A94: o = Dir o99 and
A95: p1 = Dir p1999 and
A96: q1 = Dir q1999 and
A97: not o99 is zero and
A98: not p1999 is zero and
A99: not q1999 is zero and
A100: o99,p1999,q1999 are_LinDep by A15, Th23;
A101: are_Prop q1999,q19 by A18, A21, A96, A99, ANPROJ_1:22;
A102: ( not o9,p19,p39 are_LinDep & not o9,p29,p39 are_LinDep ) by A8, A50, A51, A53, A54, A56, A57, A34, A37, Th23;
A103: p19,p29,p39 are_Prop_Vect by A56, A57, A37, Def1;
A104: are_Prop o9999,o9 by A50, A51, A83, A86, ANPROJ_1:22;
 are_Prop p3999,p39 by A34, A37, A84, A87, ANPROJ_1:22;
then A105: o9,p39,q39 are_LinDep by A89, A104, A90, ANPROJ_1:4;
A106: are_Prop o99,o9 by A50, A51, A94, A97, ANPROJ_1:22;
 are_Prop p1999,p19 by A53, A56, A95, A98, ANPROJ_1:22;
then  o9,p19,q19 are_LinDep by A100, A106, A101, ANPROJ_1:4;
then A107: o9,p19,p29,p39,q19,q29,q39 are_perspective by A71, A105, Def3;
 are_Prop r299,r29 by A44, A47, A74, A77, ANPROJ_1:22;
then  q19,q39,r29 are_LinDep by A78, A79, A82, ANPROJ_1:4;
then A108: q19,q29,q39,r19,r29,r39 lie_on_a_triangle by A81, A41, Def2;
 r19,r29,r39 are_Prop_Vect by A58, A38, A47, Def1;
then  r19,r29,r39 are_LinDep by A50, A61, A52, A60, A40, A103, A93, A107, A80, A102, A92, A108, Th8;
hence  r1,r2,r3 is_collinear by A55, A58, A35, A38, A44, A47, Th23; ::_thesis: verum
end;

Lm41:  for V being up-3-dimensional RealLinearSpace holds ProjectiveSpace V is Pappian
proof
let V be up-3-dimensional RealLinearSpace; ::_thesis: ProjectiveSpace V is Pappian
let o, p1, p2, p3, q1, q2, q3, r1, r2, r3 be Element of (ProjectiveSpace V); :: according to ANPROJ_2:def_13 ::_thesis: ( o <> p2 & o <> p3 & p2 <> p3 & p1 <> p2 & p1 <> p3 & o <> q2 & o <> q3 & q2 <> q3 & q1 <> q2 & q1 <> q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear & o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear & p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear & p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear implies r1,r2,r3 is_collinear )
assume that
A1: o <> p2 and
A2: o <> p3 and
A3: p2 <> p3 and
A4: p1 <> p2 and
A5: p1 <> p3 and
A6: o <> q2 and
A7: o <> q3 and
A8: q2 <> q3 and
A9: q1 <> q2 and
A10: q1 <> q3 and
A11: not o,p1,q1 is_collinear and
A12: o,p1,p2 is_collinear and
A13: o,p1,p3 is_collinear and
A14: o,q1,q2 is_collinear and
A15: o,q1,q3 is_collinear and
A16: p1,q2,r3 is_collinear and
A17: q1,p2,r3 is_collinear and
A18: p1,q3,r2 is_collinear and
A19: p3,q1,r2 is_collinear and
A20: p2,q3,r1 is_collinear and
A21: p3,q2,r1 is_collinear ; ::_thesis: r1,r2,r3 is_collinear
consider o999, q19, q29 being Element of V such that
A22: o = Dir o999 and
A23: q1 = Dir q19 and
A24: q2 = Dir q29 and
A25: not o999 is zero and
A26: not q19 is zero and
A27: not q29 is zero and
A28: o999,q19,q29 are_LinDep by A14, Th23;
A29: not are_Prop q19,q29 by A9, A23, A24, A26, A27, ANPROJ_1:22;
consider o9999, q199, q39 being Element of V such that
A30: o = Dir o9999 and
A31: q1 = Dir q199 and
A32: q3 = Dir q39 and
A33: not o9999 is zero and
A34: not q199 is zero and
A35: not q39 is zero and
A36: o9999,q199,q39 are_LinDep by A15, Th23;
A37: are_Prop q199,q19 by A23, A26, A31, A34, ANPROJ_1:22;
consider o99, p199, p39 being Element of V such that
A38: ( o = Dir o99 & p1 = Dir p199 ) and
A39: p3 = Dir p39 and
A40: ( not o99 is zero & not p199 is zero ) and
A41: not p39 is zero and
A42: o99,p199,p39 are_LinDep by A13, Th23;
consider o9, p19, p29 being Element of V such that
A43: o = Dir o9 and
A44: p1 = Dir p19 and
A45: p2 = Dir p29 and
A46: not o9 is zero and
A47: not p19 is zero and
A48: not p29 is zero and
A49: o9,p19,p29 are_LinDep by A12, Th23;
A50: ( not are_Prop o9,p39 & not are_Prop p19,p39 ) by A2, A5, A43, A44, A46, A47, A39, A41, ANPROJ_1:22;
A51: ( not are_Prop q19,q39 & not are_Prop q29,q39 ) by A8, A10, A23, A24, A26, A27, A32, A35, ANPROJ_1:22;
A52: not are_Prop p29,p39 by A3, A45, A48, A39, A41, ANPROJ_1:22;
A53: not are_Prop o9,q39 by A7, A43, A46, A32, A35, ANPROJ_1:22;
A54: not are_Prop o9,q29 by A6, A43, A46, A24, A27, ANPROJ_1:22;
 ( not are_Prop o9,p29 & not are_Prop p19,p29 ) by A1, A4, A43, A44, A45, A46, A47, A48, ANPROJ_1:22;
then A55: o9,p19,p29,p39,q19,q29,q39 are_half_mutually_not_Prop by A54, A53, A50, A29, A52, A51, Def5;
consider q1999, p2999, r399 being Element of V such that
A56: q1 = Dir q1999 and
A57: p2 = Dir p2999 and
A58: r3 = Dir r399 and
A59: not q1999 is zero and
A60: not p2999 is zero and
A61: not r399 is zero and
A62: q1999,p2999,r399 are_LinDep by A17, Th23;
A63: are_Prop q1999,q19 by A23, A26, A56, A59, ANPROJ_1:22;
consider p29999, q3999, r19 being Element of V such that
A64: p2 = Dir p29999 and
A65: q3 = Dir q3999 and
A66: r1 = Dir r19 and
A67: not p29999 is zero and
A68: not q3999 is zero and
A69: not r19 is zero and
A70: p29999,q3999,r19 are_LinDep by A20, Th23;
A71: are_Prop q3999,q39 by A32, A35, A65, A68, ANPROJ_1:22;
 are_Prop o999,o9 by A43, A46, A22, A25, ANPROJ_1:22;
then A72: o9,q19,q29 are_LinDep by A28, ANPROJ_1:4;
 are_Prop o9999,o9 by A43, A46, A30, A33, ANPROJ_1:22;
then A73: o9,q19,q39 are_LinDep by A36, A37, ANPROJ_1:4;
 ( are_Prop o99,o9 & are_Prop p199,p19 ) by A43, A44, A46, A47, A38, A40, ANPROJ_1:22;
then A74: o9,p19,p39 are_LinDep by A42, ANPROJ_1:4;
 not o9,p19,q19 are_LinDep by A11, A43, A44, A46, A47, A23, A26, Th23;
then A75: o9,p19,p29,p39,q19,q29,q39 lie_on_an_angle by A49, A74, A72, A73, Def4;
consider p19999, q399, r29 being Element of V such that
A76: p1 = Dir p19999 and
A77: q3 = Dir q399 and
A78: r2 = Dir r29 and
A79: not p19999 is zero and
A80: not q399 is zero and
A81: not r29 is zero and
A82: p19999,q399,r29 are_LinDep by A18, Th23;
A83: are_Prop q399,q39 by A32, A35, A77, A80, ANPROJ_1:22;
consider p3999, q29999, r199 being Element of V such that
A84: p3 = Dir p3999 and
A85: q2 = Dir q29999 and
A86: r1 = Dir r199 and
A87: not p3999 is zero and
A88: not q29999 is zero and
A89: not r199 is zero and
A90: p3999,q29999,r199 are_LinDep by A21, Th23;
A91: are_Prop p3999,p39 by A39, A41, A84, A87, ANPROJ_1:22;
A92: are_Prop q29999,q29 by A24, A27, A85, A88, ANPROJ_1:22;
 are_Prop r199,r19 by A66, A69, A86, A89, ANPROJ_1:22;
then A93: p39,q29,r19 are_LinDep by A90, A91, A92, ANPROJ_1:4;
A94: q19,q29,q39 are_Prop_Vect by A26, A27, A35, Def1;
A95: p19,p29,p39 are_Prop_Vect by A47, A48, A41, Def1;
 are_Prop p29999,p29 by A45, A48, A64, A67, ANPROJ_1:22;
then A96: p29,q39,r19 are_LinDep by A70, A71, ANPROJ_1:4;
consider p399, q1999, r299 being Element of V such that
A97: p3 = Dir p399 and
A98: q1 = Dir q1999 and
A99: r2 = Dir r299 and
A100: not p399 is zero and
A101: not q1999 is zero and
A102: not r299 is zero and
A103: p399,q1999,r299 are_LinDep by A19, Th23;
A104: are_Prop q1999,q19 by A23, A26, A98, A101, ANPROJ_1:22;
 are_Prop p19999,p19 by A44, A47, A76, A79, ANPROJ_1:22;
then A105: p19,q39,r29 are_LinDep by A82, A83, ANPROJ_1:4;
consider p1999, q2999, r39 being Element of V such that
A106: p1 = Dir p1999 and
A107: q2 = Dir q2999 and
A108: r3 = Dir r39 and
A109: not p1999 is zero and
A110: not q2999 is zero and
A111: not r39 is zero and
A112: p1999,q2999,r39 are_LinDep by A16, Th23;
A113: are_Prop q2999,q29 by A24, A27, A107, A110, ANPROJ_1:22;
A114: are_Prop p2999,p29 by A45, A48, A57, A60, ANPROJ_1:22;
 are_Prop r399,r39 by A108, A111, A58, A61, ANPROJ_1:22;
then A115: q19,p29,r39 are_LinDep by A62, A63, A114, ANPROJ_1:4;
 are_Prop p1999,p19 by A44, A47, A106, A109, ANPROJ_1:22;
then A116: p19,q29,r39 are_LinDep by A112, A113, ANPROJ_1:4;
A117: are_Prop p399,p39 by A39, A41, A97, A100, ANPROJ_1:22;
 are_Prop r299,r29 by A78, A81, A99, A102, ANPROJ_1:22;
then A118: p39,q19,r29 are_LinDep by A103, A117, A104, ANPROJ_1:4;
 r19,r29,r39 are_Prop_Vect by A111, A81, A69, Def1;
then  r19,r29,r39 are_LinDep by A46, A75, A55, A95, A94, A116, A115, A105, A118, A96, A93, Th9;
hence  r1,r2,r3 is_collinear by A108, A111, A78, A81, A66, A69, Th23; ::_thesis: verum
end;

registration
let V be up-3-dimensional RealLinearSpace;
cluster ProjectiveSpace V ->  Fanoian Desarguesian Pappian ;
coherence
 ( ProjectiveSpace V is Fanoian & ProjectiveSpace V is Desarguesian & ProjectiveSpace V is Pappian ) by Lm39, Lm40, Lm41;
end;

theorem Th27: :: ANPROJ_2:27
for V being non trivial RealLinearSpace st ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) holds
ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) implies ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) ) )
given p, q, r being Element of V such that A1: for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds
( a = 0 & b = 0 & c = 0 ) and
A2: for y being Element of V ex a, b, c being Real st y = ((a * p) + (b * q)) + (c * r) ; ::_thesis: ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) )
A3: ( not p is zero & not q is zero ) by A1, Th1;
then reconsider x1 = Dir p, x2 = Dir q as Element of (ProjectiveSpace V) by ANPROJ_1:26;
take  x1 ; ::_thesis: ex x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) )
take  x2 ; ::_thesis: ( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) )
 not are_Prop p,q by A1, Th1;
hence  x1 <> x2 by A3, ANPROJ_1:22; ::_thesis: for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear )
let r1, r2 be Element of (ProjectiveSpace V); ::_thesis: ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear )
consider u being Element of V such that
A4: ( not u is zero & r1 = Dir u ) by ANPROJ_1:26;
consider u1 being Element of V such that
A5: ( not u1 is zero & r2 = Dir u1 ) by ANPROJ_1:26;
consider y being Element of V such that
A6: p,q,y are_LinDep and
A7: u,u1,y are_LinDep and
A8: not y is zero by A1, A2, Th3;
reconsider q = Dir y as Element of (ProjectiveSpace V) by A8, ANPROJ_1:26;
take  q ; ::_thesis: ( x1,x2,q is_collinear & r1,r2,q is_collinear )
thus  x1,x2,q is_collinear by A3, A6, A8, Th23; ::_thesis: r1,r2,q is_collinear
thus  r1,r2,q is_collinear by A4, A5, A7, A8, Th23; ::_thesis: verum
end;

theorem Th28: :: ANPROJ_2:28
for V being non trivial RealLinearSpace st ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) ) holds
for p, p1, q, q1 being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) ) implies for p, p1, q, q1 being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) )
given x1, x2 being Element of (ProjectiveSpace V) such that A1: x1 <> x2 and
A2: for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ; ::_thesis: for p, p1, q, q1 being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
let p, p1, q, q1 be Element of (ProjectiveSpace V); ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
consider p3 being Element of (ProjectiveSpace V) such that
A3: x1,x2,p3 is_collinear and
A4: p,p1,p3 is_collinear by A2;
consider q3 being Element of (ProjectiveSpace V) such that
A5: x1,x2,q3 is_collinear and
A6: q,q1,q3 is_collinear by A2;
consider s2 being Element of (ProjectiveSpace V) such that
A7: x1,x2,s2 is_collinear and
A8: p,q,s2 is_collinear by A2;
A9: s2,p,q is_collinear by A8, Th24;
consider s4 being Element of (ProjectiveSpace V) such that
A10: x1,x2,s4 is_collinear and
A11: p,q1,s4 is_collinear by A2;
A12: s4,q1,p is_collinear by A11, Th24;
 p3,s2,q3 is_collinear by A1, A3, A5, A7, Def8;
then consider s3 being Element of (ProjectiveSpace V) such that
A13: p3,p,s3 is_collinear and
A14: q3,q,s3 is_collinear by A9, Def9;
consider s being Element of (ProjectiveSpace V) such that
A15: x1,x2,s is_collinear and
A16: p1,q1,s is_collinear by A2;
 q3,s4,p3 is_collinear by A1, A3, A5, A10, Def8;
then consider s5 being Element of (ProjectiveSpace V) such that
A17: q3,q1,s5 is_collinear and
A18: p3,p,s5 is_collinear by A12, Def9;
A19: p1,s,q1 is_collinear by A16, Th24;
consider s6 being Element of (ProjectiveSpace V) such that
A20: x1,x2,s6 is_collinear and
A21: p1,q,s6 is_collinear by A2;
A22: s6,p1,q is_collinear by A21, Th24;
 p3,s6,q3 is_collinear by A1, A3, A5, A20, Def8;
then consider s7 being Element of (ProjectiveSpace V) such that
A23: p3,p1,s7 is_collinear and
A24: q3,q,s7 is_collinear by A22, Def9;
 s,p3,q3 is_collinear by A1, A3, A5, A15, Def8;
then consider s1 being Element of (ProjectiveSpace V) such that
A25: p1,p3,s1 is_collinear and
A26: q1,q3,s1 is_collinear by A19, Def9;
A27: now__::_thesis:_(_p_<>_p1_&_q_<>_q1_implies_ex_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
A28: now__::_thesis:_(_p3_<>_p1_&_q3_<>_q1_implies_ex_s1,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
A29: q3,q1,s1 is_collinear by A26, Th24;
assume that
A30: p3 <> p1 and
A31: q3 <> q1 ; ::_thesis: ex s1, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 ( q3,q1,q is_collinear & q3,q1,q1 is_collinear ) by A6, Def7, Th24;
then A32: q,q1,s1 is_collinear by A31, A29, Def8;
take s1 = s1; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
A33: p3,p1,s1 is_collinear by A25, Th24;
 ( p3,p1,p is_collinear & p3,p1,p1 is_collinear ) by A4, Def7, Th24;
then  p,p1,s1 is_collinear by A30, A33, Def8;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A32; ::_thesis: verum
end;
A34: now__::_thesis:_(_p3_<>_p_&_q3_<>_q_implies_ex_s3,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
assume that
A35: p3 <> p and
A36: q3 <> q ; ::_thesis: ex s3, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
take s3 = s3; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 ( q3,q,q is_collinear & q3,q,q1 is_collinear ) by A6, Def7, Th24;
then A37: q,q1,s3 is_collinear by A14, A36, Def8;
 ( p3,p,p is_collinear & p3,p,p1 is_collinear ) by A4, Def7, Th24;
then  p,p1,s3 is_collinear by A13, A35, Def8;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A37; ::_thesis: verum
end;
A38: now__::_thesis:_(_p3_<>_p1_&_q3_<>_q_implies_ex_s7,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
assume that
A39: p3 <> p1 and
A40: q3 <> q ; ::_thesis: ex s7, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
take s7 = s7; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 ( q3,q,q is_collinear & q3,q,q1 is_collinear ) by A6, Def7, Th24;
then A41: q,q1,s7 is_collinear by A24, A40, Def8;
 ( p3,p1,p is_collinear & p3,p1,p1 is_collinear ) by A4, Def7, Th24;
then  p,p1,s7 is_collinear by A23, A39, Def8;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A41; ::_thesis: verum
end;
A42: now__::_thesis:_(_p3_<>_p_&_q3_<>_q1_implies_ex_s5,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
assume that
A43: p3 <> p and
A44: q3 <> q1 ; ::_thesis: ex s5, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
take s5 = s5; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 ( q3,q1,q is_collinear & q3,q1,q1 is_collinear ) by A6, Def7, Th24;
then A45: q,q1,s5 is_collinear by A17, A44, Def8;
 ( p3,p,p is_collinear & p3,p,p1 is_collinear ) by A4, Def7, Th24;
then  p,p1,s5 is_collinear by A18, A43, Def8;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A45; ::_thesis: verum
end;
assume  ( p <> p1 & q <> q1 ) ; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A34, A42, A38, A28; ::_thesis: verum
end;
now__::_thesis:_(_(_p_=_p1_or_q_=_q1_)_implies_ex_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
A46: now__::_thesis:_(_p_=_p1_implies_ex_q3,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
assume A47: p = p1 ; ::_thesis: ex q3, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
take q3 = q3; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 p,p1,q3 is_collinear by A47, Def7;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A6; ::_thesis: verum
end;
A48: now__::_thesis:_(_q_=_q1_implies_ex_p3,_r_being_Element_of_(ProjectiveSpace_V)_st_
(_p,p1,r_is_collinear_&_q,q1,r_is_collinear_)_)
assume A49: q = q1 ; ::_thesis: ex p3, r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
take p3 = p3; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
 q,q1,p3 is_collinear by A49, Def7;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A4; ::_thesis: verum
end;
assume  ( p = p1 or q = q1 ) ; ::_thesis: ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear )
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A48, A46; ::_thesis: verum
end;
hence  ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by A27; ::_thesis: verum
end;

theorem Th29: :: ANPROJ_2:29
for V being non trivial RealLinearSpace st ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) holds
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is 2-dimensional )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is 2-dimensional ) )
given u, v, w being Element of V such that A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) and
A2: for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ; ::_thesis: ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is 2-dimensional )
reconsider V9 = V as up-3-dimensional RealLinearSpace by A1, Def6;
take  ProjectiveSpace V9 ; ::_thesis: ( ProjectiveSpace V9 = ProjectiveSpace V & ProjectiveSpace V9 is 2-dimensional )
 ex x1, x2 being Element of (ProjectiveSpace V) st
( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st
( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) ) by A1, A2, Th27;
then  for p, p1, q, q1 being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p,p1,r is_collinear & q,q1,r is_collinear ) by Th28;
hence  ( ProjectiveSpace V9 = ProjectiveSpace V & ProjectiveSpace V9 is 2-dimensional ) by Def14; ::_thesis: verum
end;

Lm42:  for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st
for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) holds
V is up-3-dimensional
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex y, u, v, w being Element of V st
for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) implies V is up-3-dimensional )
given y, u, v, w being Element of V such that A1: for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: V is up-3-dimensional
take  y ; :: according to ANPROJ_2:def_6 ::_thesis: ex v, w1 being Element of V st
for a, b, c being Real st ((a * y) + (b * v)) + (c * w1) = 0. V holds
( a = 0 & b = 0 & c = 0 )
take  u ; ::_thesis: ex w1 being Element of V st
for a, b, c being Real st ((a * y) + (b * u)) + (c * w1) = 0. V holds
( a = 0 & b = 0 & c = 0 )
take  v ; ::_thesis: for a, b, c being Real st ((a * y) + (b * u)) + (c * v) = 0. V holds
( a = 0 & b = 0 & c = 0 )
let a, b, a1 be Real; ::_thesis: ( ((a * y) + (b * u)) + (a1 * v) = 0. V implies ( a = 0 & b = 0 & a1 = 0 ) )
assume  ((a * y) + (b * u)) + (a1 * v) = 0. V ; ::_thesis: ( a = 0 & b = 0 & a1 = 0 )
then  0. V = (((a * y) + (b * u)) + (a1 * v)) + (0. V) by RLVECT_1:4
.= (((a * y) + (b * u)) + (a1 * v)) + (0 * w) by RLVECT_1:10 ;
hence  ( a = 0 & b = 0 & a1 = 0 ) by A1; ::_thesis: verum
end;

Lm43:  for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st
for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) holds
( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex y, u, v, w being Element of V st
for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) implies ( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank ) )
given y, u, v, w being Element of V such that A1: for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: ( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank )
 V is up-3-dimensional by A1, Lm42;
hence  ( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank ) ; ::_thesis: verum
end;

theorem Th30: :: ANPROJ_2:30
for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st
( ( for w1 being Element of V ex a, b, a1, b1 being Real st w1 = (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds
ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex y, u, v, w being Element of V st
( ( for w1 being Element of V ex a, b, a1, b1 being Real st w1 = (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) implies ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) ) )
given y, u, v, w being Element of V such that A1: for w1 being Element of V ex a, b, a1, b1 being Real st w1 = (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) and
A2: for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
A3: ( not u is zero & not v is zero ) by A2, Th2;
A4: not y is zero by A2, Th2;
then reconsider p = Dir y, q1 = Dir u, q2 = Dir v as Element of (ProjectiveSpace V) by A3, ANPROJ_1:26;
take  p ; ::_thesis: ex q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
take  q1 ; ::_thesis: ex q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
take  q2 ; ::_thesis: ( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
A5: not y,u,v are_LinDep by A2, Th2;
now__::_thesis:_not_p,q1,q2_is_collinear
assume  p,q1,q2 is_collinear ; ::_thesis: contradiction
then  [p,q1,q2] in the Collinearity of (ProjectiveSpace V) by COLLSP:def_2;
hence  contradiction by A4, A3, A5, ANPROJ_1:25; ::_thesis: verum
end;
hence  not p,q1,q2 is_collinear ; ::_thesis: for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear )
let r1, r2 be Element of (ProjectiveSpace V); ::_thesis: ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear )
consider u1 being Element of V such that
A6: not u1 is zero and
A7: r1 = Dir u1 by ANPROJ_1:26;
consider u2 being Element of V such that
A8: not u2 is zero and
A9: r2 = Dir u2 by ANPROJ_1:26;
consider w1, w2 being Element of V such that
A10: u1,u2,w2 are_LinDep and
A11: u,v,w1 are_LinDep and
A12: y,w2,w1 are_LinDep and
A13: not w1 is zero and
A14: not w2 is zero by A1, A2, A6, A8, Th4;
reconsider q3 = Dir w1, r3 = Dir w2 as Element of (ProjectiveSpace V) by A13, A14, ANPROJ_1:26;
take  q3 ; ::_thesis: ex r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear )
take  r3 ; ::_thesis: ( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear )
thus  r1,r2,r3 is_collinear by A6, A7, A8, A9, A10, A14, Th23; ::_thesis: ( q1,q2,q3 is_collinear & p,r3,q3 is_collinear )
thus  q1,q2,q3 is_collinear by A3, A11, A13, Th23; ::_thesis: p,r3,q3 is_collinear
thus  p,r3,q3 is_collinear by A4, A12, A13, A14, Th23; ::_thesis: verum
end;

Lm44:  for V being non trivial RealLinearSpace
for x, d, b, b9, d9, q being Element of (ProjectiveSpace V) st not q,b,d is_collinear & b,d,x is_collinear & q,b9,b is_collinear & q,d9,d is_collinear holds
ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear )
proof
let V be non trivial RealLinearSpace; ::_thesis: for x, d, b, b9, d9, q being Element of (ProjectiveSpace V) st not q,b,d is_collinear & b,d,x is_collinear & q,b9,b is_collinear & q,d9,d is_collinear holds
ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear )
let x, d, b, b9, d9, q be Element of (ProjectiveSpace V); ::_thesis: ( not q,b,d is_collinear & b,d,x is_collinear & q,b9,b is_collinear & q,d9,d is_collinear implies ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear ) )
assume that
A1: not q,b,d is_collinear and
A2: b,d,x is_collinear and
A3: q,b9,b is_collinear and
A4: q,d9,d is_collinear ; ::_thesis: ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear )
A5: b9,q,b is_collinear by A3, Th24;
A6: b <> d by A1, Def7;
A7: now__::_thesis:_(_b_<>_b9_implies_ex_o_being_Element_of_(ProjectiveSpace_V)_st_
(_b9,d9,o_is_collinear_&_q,x,o_is_collinear_)_)
A8: b,b9,q is_collinear by A3, Th24;
consider z being Element of (ProjectiveSpace V) such that
A9: b9,d9,z is_collinear and
A10: b,d,z is_collinear by A4, A5, Def9;
A11: z,b9,b9 is_collinear by Def7;
 b,d,b is_collinear by Def7;
then  z,b,x is_collinear by A2, A6, A10, Def8;
then consider o being Element of (ProjectiveSpace V) such that
A12: z,b9,o is_collinear and
A13: x,q,o is_collinear by A8, Def9;
A14: q,x,o is_collinear by A13, Th24;
assume A15: b <> b9 ; ::_thesis: ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear )
A16: z <> b9
proof
assume  not z <> b9 ; ::_thesis: contradiction
then A17: b,b9,d is_collinear by A10, Th24;
 ( b,b9,q is_collinear & b,b9,b is_collinear ) by A3, Def7, Th24;
hence  contradiction by A1, A15, A17, Def8; ::_thesis: verum
end;
 z,b9,d9 is_collinear by A9, Th24;
then  b9,d9,o is_collinear by A12, A16, A11, Def8;
hence  ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear ) by A14; ::_thesis: verum
end;
now__::_thesis:_(_b_=_b9_implies_ex_o_being_Element_of_(ProjectiveSpace_V)_st_
(_b9,d9,o_is_collinear_&_q,x,o_is_collinear_)_)
assume  b = b9 ; ::_thesis: ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear )
then A18: d,b9,x is_collinear by A2, Th24;
 d9,d,q is_collinear by A4, Th24;
then consider o being Element of (ProjectiveSpace V) such that
A19: d9,b9,o is_collinear and
A20: q,x,o is_collinear by A18, Def9;
 b9,d9,o is_collinear by A19, Th24;
hence  ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear ) by A20; ::_thesis: verum
end;
hence  ex o being Element of (ProjectiveSpace V) st
( b9,d9,o is_collinear & q,x,o is_collinear ) by A7; ::_thesis: verum
end;


theorem Th31: :: ANPROJ_2:31
for V being non trivial RealLinearSpace st ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) ) holds
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) )
assume that
A1: ProjectiveSpace V is proper and
A2: for p, q being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st
( p <> r & q <> r & p,q,r is_collinear ) ; :: according to ANPROJ_2:def_10 ::_thesis: ( for p, q1, q2 being Element of (ProjectiveSpace V) holds
( p,q1,q2 is_collinear or ex r1, r2 being Element of (ProjectiveSpace V) st
for q3, r3 being Element of (ProjectiveSpace V) holds
( not r1,r2,r3 is_collinear or not q1,q2,q3 is_collinear or not p,r3,q3 is_collinear ) ) or ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) )
defpred S1[ Element of (ProjectiveSpace V), Element of (ProjectiveSpace V), Element of (ProjectiveSpace V)] means  for y1, y2 being Element of (ProjectiveSpace V) ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & $2,$3,x2 is_collinear & $1,x1,x2 is_collinear );
A3: for p, q1, q2 being Element of (ProjectiveSpace V) st q1,q2,p is_collinear holds
S1[p,q1,q2]
proof
let p, q1, q2 be Element of (ProjectiveSpace V); ::_thesis: ( q1,q2,p is_collinear implies S1[p,q1,q2] )
assume A4: q1,q2,p is_collinear ; ::_thesis: S1[p,q1,q2]
now__::_thesis:_for_y1,_y2_being_Element_of_(ProjectiveSpace_V)_ex_x2,_x1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,x1_is_collinear_&_q1,q2,x2_is_collinear_&_p,x1,x2_is_collinear_)
let y1, y2 be Element of (ProjectiveSpace V); ::_thesis: ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & p,x1,x2 is_collinear )
 ( y1,y2,y2 is_collinear & p,y2,p is_collinear ) by Def7;
hence  ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & p,x1,x2 is_collinear ) by A4; ::_thesis: verum
end;
hence  S1[p,q1,q2] ; ::_thesis: verum
end;
A5: for q, q1, q2, p1, p2, x being Element of (ProjectiveSpace V) st S1[q,q1,q2] & not q1,q2,q is_collinear & q1,q2,x is_collinear & not p1,p2,q is_collinear & p1,p2,x is_collinear holds
S1[q,p1,p2]
proof
let q, q1, q2, p1, p2, x be Element of (ProjectiveSpace V); ::_thesis: ( S1[q,q1,q2] & not q1,q2,q is_collinear & q1,q2,x is_collinear & not p1,p2,q is_collinear & p1,p2,x is_collinear implies S1[q,p1,p2] )
assume that
A6: S1[q,q1,q2] and
A7: not q1,q2,q is_collinear and
A8: q1,q2,x is_collinear and
A9: not p1,p2,q is_collinear and
A10: p1,p2,x is_collinear ; ::_thesis: S1[q,p1,p2]
A11: q1 <> q2 by A7, Def7;
A12: p1 <> p2 by A9, Def7;
now__::_thesis:_for_y1,_y2_being_Element_of_(ProjectiveSpace_V)_ex_z2,_z1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,z1_is_collinear_&_p1,p2,z2_is_collinear_&_q,z1,z2_is_collinear_)
let y1, y2 be Element of (ProjectiveSpace V); ::_thesis: ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear )
A13: now__::_thesis:_(_y1_<>_y2_implies_ex_z2,_z1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,z1_is_collinear_&_p1,p2,z2_is_collinear_&_q,z1,z2_is_collinear_)_)
 ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a )
proof
A14: now__::_thesis:_(_x_<>_p2_implies_ex_p2,_a_being_Element_of_(ProjectiveSpace_V)_st_
(_p1,p2,a_is_collinear_&_x_<>_a_)_)
assume A15: x <> p2 ; ::_thesis: ex p2, a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a )
take p2 = p2; ::_thesis: ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a )
 p1,p2,p2 is_collinear by Def7;
hence  ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a ) by A15; ::_thesis: verum
end;
now__::_thesis:_(_x_<>_p1_implies_ex_p1,_a_being_Element_of_(ProjectiveSpace_V)_st_
(_p1,p2,a_is_collinear_&_x_<>_a_)_)
assume A16: x <> p1 ; ::_thesis: ex p1, a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a )
take p1 = p1; ::_thesis: ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a )
 p1,p2,p1 is_collinear by Def7;
hence  ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a ) by A16; ::_thesis: verum
end;
hence  ex a being Element of (ProjectiveSpace V) st
( p1,p2,a is_collinear & x <> a ) by A9, A14, Def7; ::_thesis: verum
end;
then consider x1 being Element of (ProjectiveSpace V) such that
A17: p1,p2,x1 is_collinear and
A18: x <> x1 ;
consider b, b9 being Element of (ProjectiveSpace V) such that
A19: y1,y2,b9 is_collinear and
A20: q1,q2,b is_collinear and
A21: q,b9,b is_collinear by A6;
assume A22: y1 <> y2 ; ::_thesis: ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear )
 ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a )
proof
A23: now__::_thesis:_(_b9_<>_y2_implies_ex_y2,_a_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,a_is_collinear_&_b9_<>_a_)_)
assume A24: b9 <> y2 ; ::_thesis: ex y2, a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a )
take y2 = y2; ::_thesis: ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a )
 y1,y2,y2 is_collinear by Def7;
hence  ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a ) by A24; ::_thesis: verum
end;
now__::_thesis:_(_b9_<>_y1_implies_ex_y1,_a_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,a_is_collinear_&_b9_<>_a_)_)
assume A25: b9 <> y1 ; ::_thesis: ex y1, a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a )
take y1 = y1; ::_thesis: ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a )
 y1,y2,y1 is_collinear by Def7;
hence  ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a ) by A25; ::_thesis: verum
end;
hence  ex a being Element of (ProjectiveSpace V) st
( y1,y2,a is_collinear & b9 <> a ) by A22, A23; ::_thesis: verum
end;
then consider x3 being Element of (ProjectiveSpace V) such that
A26: b9 <> x3 and
A27: y1,y2,x3 is_collinear ;
consider d, d9 being Element of (ProjectiveSpace V) such that
A28: x1,x3,d9 is_collinear and
A29: q1,q2,d is_collinear and
A30: q,d9,d is_collinear by A6;
A31: b,d,x is_collinear by A8, A11, A20, A29, Def8;
A32: now__::_thesis:_(_b_<>_d_implies_ex_z2,_z1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,z1_is_collinear_&_p1,p2,z2_is_collinear_&_q,z1,z2_is_collinear_)_)
assume A33: b <> d ; ::_thesis: ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear )
 not q,b,d is_collinear
proof
 q1,q2,q2 is_collinear by Def7;
then A34: b,d,q2 is_collinear by A11, A20, A29, Def8;
assume  q,b,d is_collinear ; ::_thesis: contradiction
then A35: b,d,q is_collinear by Th24;
 q1,q2,q1 is_collinear by Def7;
then  b,d,q1 is_collinear by A11, A20, A29, Def8;
hence  contradiction by A7, A33, A35, A34, Def8; ::_thesis: verum
end;
then consider o being Element of (ProjectiveSpace V) such that
A36: b9,d9,o is_collinear and
A37: q,x,o is_collinear by A21, A30, A31, Lm44;
A38: o,x,q is_collinear by A37, Th24;
 d9,x3,x1 is_collinear by A28, Th24;
then consider z1 being Element of (ProjectiveSpace V) such that
A39: b9,x3,z1 is_collinear and
A40: o,x1,z1 is_collinear by A36, Def9;
 x1,o,z1 is_collinear by A40, Th24;
then consider z2 being Element of (ProjectiveSpace V) such that
A41: x1,x,z2 is_collinear and
A42: z1,q,z2 is_collinear by A38, Def9;
A43: q,z1,z2 is_collinear by A42, Th24;
 p1,p2,p2 is_collinear by Def7;
then A44: x1,x,p2 is_collinear by A10, A12, A17, Def8;
 y1,y2,y2 is_collinear by Def7;
then A45: b9,x3,y2 is_collinear by A22, A19, A27, Def8;
 p1,p2,p1 is_collinear by Def7;
then  x1,x,p1 is_collinear by A10, A12, A17, Def8;
then A46: p1,p2,z2 is_collinear by A18, A41, A44, Def8;
 y1,y2,y1 is_collinear by Def7;
then  b9,x3,y1 is_collinear by A22, A19, A27, Def8;
then  y1,y2,z1 is_collinear by A26, A39, A45, Def8;
hence  ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear ) by A46, A43; ::_thesis: verum
end;
now__::_thesis:_(_b_=_d_implies_ex_z2,_z1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,z1_is_collinear_&_p1,p2,z2_is_collinear_&_q,z1,z2_is_collinear_)_)
assume  b = d ; ::_thesis: ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear )
then A47: b,q,d9 is_collinear by A30, Th24;
 y1,y2,y2 is_collinear by Def7;
then A48: b9,x3,y2 is_collinear by A22, A19, A27, Def8;
A49: d9,x3,x1 is_collinear by A28, Th24;
 ( b,q,b9 is_collinear & b,q,q is_collinear ) by A21, Def7, Th24;
then  b9,d9,q is_collinear by A7, A20, A47, Def8;
then consider z1 being Element of (ProjectiveSpace V) such that
A50: b9,x3,z1 is_collinear and
A51: q,x1,z1 is_collinear by A49, Def9;
A52: q,z1,x1 is_collinear by A51, Th24;
 y1,y2,y1 is_collinear by Def7;
then  b9,x3,y1 is_collinear by A22, A19, A27, Def8;
then  y1,y2,z1 is_collinear by A26, A50, A48, Def8;
hence  ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear ) by A17, A52; ::_thesis: verum
end;
hence  ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear ) by A32; ::_thesis: verum
end;
now__::_thesis:_(_y1_=_y2_implies_ex_z2,_z1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,z1_is_collinear_&_p1,p2,z2_is_collinear_&_q,z1,z2_is_collinear_)_)
assume  y1 = y2 ; ::_thesis: ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear )
then A53: y1,y2,q is_collinear by Def7;
 ( p1,p2,p1 is_collinear & q,q,p1 is_collinear ) by Def7;
hence  ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear ) by A53; ::_thesis: verum
end;
hence  ex z2, z1 being Element of (ProjectiveSpace V) st
( y1,y2,z1 is_collinear & p1,p2,z2 is_collinear & q,z1,z2 is_collinear ) by A13; ::_thesis: verum
end;
hence  S1[q,p1,p2] ; ::_thesis: verum
end;
A54: for q1, q2, p1, p2, q being Element of (ProjectiveSpace V) st not q1,q2,q is_collinear & not p1,p2,q is_collinear & ( for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ) holds
ex q3, p3 being Element of (ProjectiveSpace V) st
( p1,p2,p3 is_collinear & q1,q2,q3 is_collinear & not q3,p3,q is_collinear )
proof
let q1, q2, p1, p2, q be Element of (ProjectiveSpace V); ::_thesis: ( not q1,q2,q is_collinear & not p1,p2,q is_collinear & ( for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ) implies ex q3, p3 being Element of (ProjectiveSpace V) st
( p1,p2,p3 is_collinear & q1,q2,q3 is_collinear & not q3,p3,q is_collinear ) )
assume that
A55: not q1,q2,q is_collinear and
A56: not p1,p2,q is_collinear and
 for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ; ::_thesis: ex q3, p3 being Element of (ProjectiveSpace V) st
( p1,p2,p3 is_collinear & q1,q2,q3 is_collinear & not q3,p3,q is_collinear )
A57: q <> q1 by A55, Def7;
A58: ( not q1,p1,q is_collinear or not q1,p2,q is_collinear )
proof
assume  ( q1,p1,q is_collinear & q1,p2,q is_collinear ) ; ::_thesis: contradiction
then A59: ( q,q1,p1 is_collinear & q,q1,p2 is_collinear ) by Th24;
 q,q1,q is_collinear by Def7;
hence  contradiction by A56, A57, A59, Def8; ::_thesis: verum
end;
A60: p1,p2,p2 is_collinear by Def7;
 ( q1,q2,q1 is_collinear & p1,p2,p1 is_collinear ) by Def7;
hence  ex q3, p3 being Element of (ProjectiveSpace V) st
( p1,p2,p3 is_collinear & q1,q2,q3 is_collinear & not q3,p3,q is_collinear ) by A60, A58; ::_thesis: verum
end;
A61: for q, q1, q2, p1, p2 being Element of (ProjectiveSpace V) st S1[q,q1,q2] & not q1,q2,q is_collinear & not p1,p2,q is_collinear & ( for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ) holds
S1[q,p1,p2]
proof
let q, q1, q2, p1, p2 be Element of (ProjectiveSpace V); ::_thesis: ( S1[q,q1,q2] & not q1,q2,q is_collinear & not p1,p2,q is_collinear & ( for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ) implies S1[q,p1,p2] )
assume that
A62: S1[q,q1,q2] and
A63: not q1,q2,q is_collinear and
A64: not p1,p2,q is_collinear and
A65: for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ; ::_thesis: S1[q,p1,p2]
consider q3, p3 being Element of (ProjectiveSpace V) such that
A66: p1,p2,p3 is_collinear and
A67: q1,q2,q3 is_collinear and
A68: not q3,p3,q is_collinear by A54, A63, A64, A65;
 q3,p3,q3 is_collinear by Def7;
then A69: S1[q,q3,p3] by A5, A62, A63, A67, A68;
 q3,p3,p3 is_collinear by Def7;
hence  S1[q,p1,p2] by A5, A64, A66, A68, A69; ::_thesis: verum
end;
A70: for q, q1, q2 being Element of (ProjectiveSpace V) st S1[q,q1,q2] & not q1,q2,q is_collinear holds
for p1, p2 being Element of (ProjectiveSpace V) holds S1[q,p1,p2]
proof
let q, q1, q2 be Element of (ProjectiveSpace V); ::_thesis: ( S1[q,q1,q2] & not q1,q2,q is_collinear implies for p1, p2 being Element of (ProjectiveSpace V) holds S1[q,p1,p2] )
assume A71: ( S1[q,q1,q2] & not q1,q2,q is_collinear ) ; ::_thesis: for p1, p2 being Element of (ProjectiveSpace V) holds S1[q,p1,p2]
let p1, p2 be Element of (ProjectiveSpace V); ::_thesis: S1[q,p1,p2]
A72: ( not p1,p2,q is_collinear & ( for x being Element of (ProjectiveSpace V) holds
( not q1,q2,x is_collinear or not p1,p2,x is_collinear ) ) implies S1[q,p1,p2] ) by A61, A71;
 ( not p1,p2,q is_collinear & ex x being Element of (ProjectiveSpace V) st
( q1,q2,x is_collinear & p1,p2,x is_collinear ) implies S1[q,p1,p2] ) by A5, A71;
hence  S1[q,p1,p2] by A3, A72; ::_thesis: verum
end;
reconsider CS = ProjectiveSpace V as CollProjectiveSpace by A1, A2, Def10;
given p, q1, q2 being Element of (ProjectiveSpace V) such that A73: not p,q1,q2 is_collinear and
A74: for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ; ::_thesis: ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
take  CS ; ::_thesis: ( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
A75: for q, q1, q2, x, q3 being Element of (ProjectiveSpace V) st S1[q,q1,q2] & not q1,q2,q is_collinear & q1,q2,x is_collinear & q,q3,x is_collinear holds
S1[q3,q1,q2]
proof
let q, q1, q2, x, q3 be Element of (ProjectiveSpace V); ::_thesis: ( S1[q,q1,q2] & not q1,q2,q is_collinear & q1,q2,x is_collinear & q,q3,x is_collinear implies S1[q3,q1,q2] )
assume that
A76: S1[q,q1,q2] and
A77: not q1,q2,q is_collinear and
A78: q1,q2,x is_collinear and
A79: q,q3,x is_collinear ; ::_thesis: S1[q3,q1,q2]
now__::_thesis:_for_y1,_y2_being_Element_of_(ProjectiveSpace_V)_ex_x2,_x1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,x1_is_collinear_&_q1,q2,x2_is_collinear_&_q3,x1,x2_is_collinear_)
let y1, y2 be Element of (ProjectiveSpace V); ::_thesis: ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear )
consider z2, z1 being Element of (ProjectiveSpace V) such that
A80: y1,y2,z1 is_collinear and
A81: q1,q2,z2 is_collinear and
A82: q,z1,z2 is_collinear by A76;
A83: now__::_thesis:_(_x_<>_z2_implies_ex_x2,_x1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,x1_is_collinear_&_q1,q2,x2_is_collinear_&_q3,x1,x2_is_collinear_)_)
 q3,q,x is_collinear by A79, Th24;
then consider x2 being Element of (ProjectiveSpace V) such that
A84: q3,z1,x2 is_collinear and
A85: x,z2,x2 is_collinear by A82, Def9;
A86: q1 <> q2 by A77, Def7;
 q1,q2,q2 is_collinear by Def7;
then A87: x,z2,q2 is_collinear by A78, A81, A86, Def8;
 q1,q2,q1 is_collinear by Def7;
then A88: x,z2,q1 is_collinear by A78, A81, A86, Def8;
assume  x <> z2 ; ::_thesis: ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear )
then  q1,q2,x2 is_collinear by A85, A88, A87, Def8;
hence  ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear ) by A80, A84; ::_thesis: verum
end;
now__::_thesis:_(_x_=_z2_implies_ex_x2,_x1_being_Element_of_(ProjectiveSpace_V)_st_
(_y1,y2,x1_is_collinear_&_q1,q2,x2_is_collinear_&_q3,x1,x2_is_collinear_)_)
A89: ( q,x,q3 is_collinear & q,x,x is_collinear ) by A79, Def7, Th24;
assume A90: x = z2 ; ::_thesis: ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear )
then  q,x,z1 is_collinear by A82, Th24;
then  q3,z1,z2 is_collinear by A77, A78, A90, A89, Def8;
hence  ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear ) by A80, A81; ::_thesis: verum
end;
hence  ex x2, x1 being Element of (ProjectiveSpace V) st
( y1,y2,x1 is_collinear & q1,q2,x2 is_collinear & q3,x1,x2 is_collinear ) by A83; ::_thesis: verum
end;
hence  S1[q3,q1,q2] ; ::_thesis: verum
end;
A91: for q, p being Element of (ProjectiveSpace V) st ( for q1, q2 being Element of (ProjectiveSpace V) holds S1[q,q1,q2] ) holds
ex p1, p2 being Element of (ProjectiveSpace V) st
( S1[p,p1,p2] & not p1,p2,p is_collinear )
proof
let q, p be Element of (ProjectiveSpace V); ::_thesis: ( ( for q1, q2 being Element of (ProjectiveSpace V) holds S1[q,q1,q2] ) implies ex p1, p2 being Element of (ProjectiveSpace V) st
( S1[p,p1,p2] & not p1,p2,p is_collinear ) )
assume A92: for q1, q2 being Element of (ProjectiveSpace V) holds S1[q,q1,q2] ; ::_thesis: ex p1, p2 being Element of (ProjectiveSpace V) st
( S1[p,p1,p2] & not p1,p2,p is_collinear )
consider x1 being Element of (ProjectiveSpace V) such that
A93: p <> x1 and
A94: q <> x1 and
A95: p,q,x1 is_collinear by A2;
consider x2 being Element of (ProjectiveSpace V) such that
A96: not p,x1,x2 is_collinear by A1, A93, COLLSP:12;
A97: not x1,x2,q is_collinear
proof
assume  x1,x2,q is_collinear ; ::_thesis: contradiction
then A98: q,x1,x2 is_collinear by Th24;
 ( q,x1,x1 is_collinear & q,x1,p is_collinear ) by A95, Def7, Th24;
hence  contradiction by A94, A96, A98, Def8; ::_thesis: verum
end;
A99: x1,x2,x1 is_collinear by Def7;
A100: not x1,x2,p is_collinear by A96, Th24;
A101: S1[q,x1,x2] by A92;
 q,p,x1 is_collinear by A95, Th24;
then  S1[p,x1,x2] by A75, A97, A99, A101;
hence  ex p1, p2 being Element of (ProjectiveSpace V) st
( S1[p,p1,p2] & not p1,p2,p is_collinear ) by A100; ::_thesis: verum
end;
A102: for x, y1, z being Element of (ProjectiveSpace V) holds S1[x,y1,z]
proof
let x, y1, z be Element of (ProjectiveSpace V); ::_thesis: S1[x,y1,z]
 not q1,q2,p is_collinear by A73, Th24;
then  for p1, p2 being Element of (ProjectiveSpace V) holds S1[p,p1,p2] by A74, A70;
then  ex r1, r2 being Element of (ProjectiveSpace V) st
( S1[x,r1,r2] & not r1,r2,x is_collinear ) by A91;
hence  S1[x,y1,z] by A70; ::_thesis: verum
end;
 for p4, p1, q, q4, r2 being Element of (ProjectiveSpace V) ex r, r1 being Element of (ProjectiveSpace V) st
( p4,q,r is_collinear & p1,q4,r1 is_collinear & r2,r,r1 is_collinear )
proof
let p4, p1, q, q4, r2 be Element of (ProjectiveSpace V); ::_thesis: ex r, r1 being Element of (ProjectiveSpace V) st
( p4,q,r is_collinear & p1,q4,r1 is_collinear & r2,r,r1 is_collinear )
 ex r1, r being Element of (ProjectiveSpace V) st
( p4,q,r is_collinear & p1,q4,r1 is_collinear & r2,r,r1 is_collinear ) by A102;
hence  ex r, r1 being Element of (ProjectiveSpace V) st
( p4,q,r is_collinear & p1,q4,r1 is_collinear & r2,r,r1 is_collinear ) ; ::_thesis: verum
end;
hence  ( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) by Def15; ::_thesis: verum
end;

theorem Th32: :: ANPROJ_2:32
for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st
( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex y, u, v, w being Element of V st
( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) )
given y, u, v, w being Element of V such that A1: ( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) ; ::_thesis: ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
 ( ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st
( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st
( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) ) ) by A1, Lm43, Th30;
hence  ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is at_most-3-dimensional ) by Th31; ::_thesis: verum
end;

theorem Th33: :: ANPROJ_2:33
for V being non trivial RealLinearSpace st ex u, v, u1, v1 being Element of V st
for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) holds
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex u, v, u1, v1 being Element of V st
for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional ) )
given u, v, u1, v1 being Element of V such that A1: for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ; ::_thesis: ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & not CS is 2-dimensional )
 V is up-3-dimensional by A1, Lm42;
then reconsider CS = ProjectiveSpace V as CollProjectiveSpace ;
take  CS ; ::_thesis: ( CS = ProjectiveSpace V & not CS is 2-dimensional )
thus  CS = ProjectiveSpace V ; ::_thesis: not CS is 2-dimensional
A2: ( not u1 is zero & not v1 is zero ) by A1, Th2;
A3: ( not u is zero & not v is zero ) by A1, Th2;
then reconsider p = Dir u, p1 = Dir v, q = Dir u1, q1 = Dir v1 as Element of CS by A2, ANPROJ_1:26;
take  p ; :: according to ANPROJ_2:def_14 ::_thesis: ex p1, q, q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take  p1 ; ::_thesis: ex q, q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take  q ; ::_thesis: ex q1 being Element of CS st
for r being Element of CS holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
take  q1 ; ::_thesis: for r being Element of CS holds
( not p,p1,r is_collinear or not q,q1,r is_collinear )
thus  for r being Element of CS holds
( not p,p1,r is_collinear or not q,q1,r is_collinear ) ::_thesis: verum
proof
assume  ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ; ::_thesis: contradiction
then consider r being Element of CS such that
A4: p,p1,r is_collinear and
A5: q,q1,r is_collinear ;
consider y being Element of V such that
A6: not y is zero and
A7: r = Dir y by ANPROJ_1:26;
 [q,q1,r] in the Collinearity of (ProjectiveSpace V) by A5, COLLSP:def_2;
then A8: u1,v1,y are_LinDep by A2, A6, A7, ANPROJ_1:25;
 [p,p1,r] in the Collinearity of (ProjectiveSpace V) by A4, COLLSP:def_2;
then  u,v,y are_LinDep by A3, A6, A7, ANPROJ_1:25;
hence  contradiction by A1, A6, A8, Th5; ::_thesis: verum
end;
end;

theorem Th34: :: ANPROJ_2:34
for V being non trivial RealLinearSpace st ex u, v, u1, v1 being Element of V st
( ( for w being Element of V ex a, b, a1, b1 being Real st w = (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) ) & ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds
ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional )
proof
let V be non trivial RealLinearSpace; ::_thesis: ( ex u, v, u1, v1 being Element of V st
( ( for w being Element of V ex a, b, a1, b1 being Real st w = (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) ) & ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) implies ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional ) )
assume  ex u, v, u1, v1 being Element of V st
( ( for w being Element of V ex a, b, a1, b1 being Real st w = (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) ) & ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) ; ::_thesis: ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional )
then  ( ex CS1 being CollProjectiveSpace st
( CS1 = ProjectiveSpace V & CS1 is up-3-dimensional ) & ex CS2 being CollProjectiveSpace st
( CS2 = ProjectiveSpace V & CS2 is at_most-3-dimensional ) ) by Th32, Th33;
hence  ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional ) ; ::_thesis: verum
end;

registration
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian 2-dimensional for CollStr ;
existence
 ex b1 being CollProjectiveSpace st
( b1 is strict & b1 is Fanoian & b1 is Desarguesian & b1 is Pappian & b1 is 2-dimensional )
proof
consider V being non trivial RealLinearSpace such that
A1: ex u, v, w being Element of V st
( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) by Th16;
reconsider V = V as up-3-dimensional RealLinearSpace by A1, Def6;
take CS = ProjectiveSpace V; ::_thesis: ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian & CS is 2-dimensional )
thus  ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian ) ; ::_thesis: CS is 2-dimensional
 ex CS1 being CollProjectiveSpace st
( CS1 = ProjectiveSpace V & CS1 is 2-dimensional ) by A1, Th29;
hence  CS is 2-dimensional ; ::_thesis: verum
end;
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian up-3-dimensional at_most-3-dimensional for CollStr ;
existence
 ex b1 being CollProjectiveSpace st
( b1 is strict & b1 is Fanoian & b1 is Desarguesian & b1 is Pappian & b1 is at_most-3-dimensional & b1 is up-3-dimensional )
proof
consider V being non trivial RealLinearSpace such that
A2: ex u, v, w, u1 being Element of V st
( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds
( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) ) by Th22;
reconsider V = V as up-3-dimensional RealLinearSpace by A2, Lm42;
take CS = ProjectiveSpace V; ::_thesis: ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian & CS is at_most-3-dimensional & CS is up-3-dimensional )
thus  ( CS is strict & CS is Fanoian & CS is Desarguesian & CS is Pappian ) ; ::_thesis: ( CS is at_most-3-dimensional & CS is up-3-dimensional )
 ex CS being CollProjectiveSpace st
( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional ) by A2, Th34;
hence  ( CS is at_most-3-dimensional & CS is up-3-dimensional ) ; ::_thesis: verum
end;
end;

definition
mode CollProjectivePlane is 2-dimensional CollProjectiveSpace;
end;

theorem :: ANPROJ_2:35
for CS being non empty CollStr holds
( CS is 2-dimensional CollProjectiveSpace iff ( CS is proper at_least_3rank CollSp & ( for p, p1, q, q1 being Element of CS ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ) ) )
proof
let CS be non empty CollStr ; ::_thesis: ( CS is 2-dimensional CollProjectiveSpace iff ( CS is proper at_least_3rank CollSp & ( for p, p1, q, q1 being Element of CS ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ) ) )
thus  ( CS is 2-dimensional CollProjectiveSpace implies ( CS is proper at_least_3rank CollSp & ( for p, p1, q, q1 being Element of CS ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ) ) ) by Def14; ::_thesis: ( CS is proper at_least_3rank CollSp & ( for p, p1, q, q1 being Element of CS ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ) implies CS is 2-dimensional CollProjectiveSpace )
assume that
A1: CS is proper at_least_3rank CollSp and
A2: for p, p1, q, q1 being Element of CS ex r being Element of CS st
( p,p1,r is_collinear & q,q1,r is_collinear ) ; ::_thesis: CS is 2-dimensional CollProjectiveSpace
 CS is Vebleian
proof
let p, p1, p2, r, r1 be Element of CS; :: according to ANPROJ_2:def_9 ::_thesis: ( p,p1,r is_collinear & p1,p2,r1 is_collinear implies ex r2 being Element of CS st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) )
assume that
 p,p1,r is_collinear and
 p1,p2,r1 is_collinear ; ::_thesis: ex r2 being Element of CS st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear )
thus  ex r2 being Element of CS st
( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) by A2; ::_thesis: verum
end;
hence  CS is 2-dimensional CollProjectiveSpace by A1, A2, Def14; ::_thesis: verum
end;