:: COMBGRAS semantic presentation
begin
theorem Th1: :: COMBGRAS:1
for n being Element of NAT
for a, b being set st a <> b & card a = n & card b = n holds
( card (a /\ b) in n & n + 1 c= card (a \/ b) )
proof
let n be Element of NAT ; ::_thesis: for a, b being set st a <> b & card a = n & card b = n holds
( card (a /\ b) in n & n + 1 c= card (a \/ b) )
let a, b be set ; ::_thesis: ( a <> b & card a = n & card b = n implies ( card (a /\ b) in n & n + 1 c= card (a \/ b) ) )
assume that
A1: a <> b and
A2: card a = n and
A3: card b = n and
A4: ( not card (a /\ b) in n or not n + 1 c= card (a \/ b) ) ; ::_thesis: contradiction
( n c= card (a /\ b) or card (a \/ b) in n + 1 ) by A4, ORDINAL1:16;
then ( n c= card (a /\ b) or card (a \/ b) in succ n ) by NAT_1:38;
then A5: ( n c= card (a /\ b) or card (a \/ b) c= n ) by ORDINAL1:22;
( n c= card (a \/ b) & card (a /\ b) c= n ) by A2, CARD_1:11, XBOOLE_1:7, XBOOLE_1:17;
then A6: ( ( card a = card (a /\ b) & card (a /\ b) = card b ) or ( card (a \/ b) = card a & card (a \/ b) = card b ) ) by A2, A3, A5, XBOOLE_0:def_10;
A7: ( a c= a \/ b & b c= a \/ b ) by XBOOLE_1:7;
( a is finite set & b is finite set ) by A2, A3;
then ( ( a = a /\ b & b = a /\ b ) or ( a = a \/ b & b = a \/ b ) ) by A7, A6, CARD_FIN:1, XBOOLE_1:17;
hence contradiction by A1; ::_thesis: verum
end;
theorem Th2: :: COMBGRAS:2
for n, k being Element of NAT
for a, b being set st card a = n + k & card b = n + k holds
( card (a /\ b) = n iff card (a \/ b) = n + (2 * k) )
proof
let n, k be Element of NAT ; ::_thesis: for a, b being set st card a = n + k & card b = n + k holds
( card (a /\ b) = n iff card (a \/ b) = n + (2 * k) )
let a, b be set ; ::_thesis: ( card a = n + k & card b = n + k implies ( card (a /\ b) = n iff card (a \/ b) = n + (2 * k) ) )
assume that
A1: card a = n + k and
A2: card b = n + k ; ::_thesis: ( card (a /\ b) = n iff card (a \/ b) = n + (2 * k) )
A3: a is finite by A1;
A4: b is finite by A2;
thus ( card (a /\ b) = n implies card (a \/ b) = n + (2 * k) ) ::_thesis: ( card (a \/ b) = n + (2 * k) implies card (a /\ b) = n )
proof
assume card (a /\ b) = n ; ::_thesis: card (a \/ b) = n + (2 * k)
then card (a \/ b) = ((n + k) + (n + k)) - n by A1, A2, A3, A4, CARD_2:45;
hence card (a \/ b) = n + (2 * k) ; ::_thesis: verum
end;
thus ( card (a \/ b) = n + (2 * k) implies card (a /\ b) = n ) ::_thesis: verum
proof
reconsider m = card (a /\ b) as Nat by A3;
assume card (a \/ b) = n + (2 * k) ; ::_thesis: card (a /\ b) = n
then n + (2 * k) = ((n + k) + (n + k)) - m by A1, A2, A3, A4, CARD_2:45;
hence card (a /\ b) = n ; ::_thesis: verum
end;
end;
theorem Th3: :: COMBGRAS:3
for X, Y being set holds
( card X c= card Y iff ex f being Function st
( f is one-to-one & X c= dom f & f .: X c= Y ) )
proof
let X, Y be set ; ::_thesis: ( card X c= card Y iff ex f being Function st
( f is one-to-one & X c= dom f & f .: X c= Y ) )
thus ( card X c= card Y implies ex f being Function st
( f is one-to-one & X c= dom f & f .: X c= Y ) ) ::_thesis: ( ex f being Function st
( f is one-to-one & X c= dom f & f .: X c= Y ) implies card X c= card Y )
proof
assume card X c= card Y ; ::_thesis: ex f being Function st
( f is one-to-one & X c= dom f & f .: X c= Y )
then consider f being Function such that
A1: ( f is one-to-one & dom f = X & rng f c= Y ) by CARD_1:10;
take f ; ::_thesis: ( f is one-to-one & X c= dom f & f .: X c= Y )
thus ( f is one-to-one & X c= dom f & f .: X c= Y ) by A1, RELAT_1:113; ::_thesis: verum
end;
given f being Function such that A2: f is one-to-one and
A3: X c= dom f and
A4: f .: X c= Y ; ::_thesis: card X c= card Y
A5: rng (f | X) c= Y
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f | X) or y in Y )
assume y in rng (f | X) ; ::_thesis: y in Y
then consider x being set such that
A6: ( x in dom (f | X) & y = (f | X) . x ) by FUNCT_1:def_3;
( x in X & y = f . x ) by A3, A6, FUNCT_1:47, RELAT_1:62;
then y in f .: X by A3, FUNCT_1:def_6;
hence y in Y by A4; ::_thesis: verum
end;
( f | X is one-to-one & dom (f | X) = X ) by A2, A3, FUNCT_1:52, RELAT_1:62;
hence card X c= card Y by A5, CARD_1:10; ::_thesis: verum
end;
theorem Th4: :: COMBGRAS:4
for X being set
for f being Function st f is one-to-one & X c= dom f holds
card (f .: X) = card X
proof
let X be set ; ::_thesis: for f being Function st f is one-to-one & X c= dom f holds
card (f .: X) = card X
let f be Function; ::_thesis: ( f is one-to-one & X c= dom f implies card (f .: X) = card X )
assume ( f is one-to-one & X c= dom f ) ; ::_thesis: card (f .: X) = card X
then ( card (f .: X) c= card X & card X c= card (f .: X) ) by Th3, CARD_1:67;
hence card (f .: X) = card X by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th5: :: COMBGRAS:5
for X, Y, Z being set st X \ Y = X \ Z & Y c= X & Z c= X holds
Y = Z
proof
let X, Y, Z be set ; ::_thesis: ( X \ Y = X \ Z & Y c= X & Z c= X implies Y = Z )
assume that
A1: X \ Y = X \ Z and
A2: Y c= X and
A3: Z c= X ; ::_thesis: Y = Z
Z \/ X = X by A3, XBOOLE_1:12;
then A4: Y \ Z c= X by A2, XBOOLE_1:43;
X \ Z misses Y by A1, XBOOLE_1:106;
then Y \ Z = {} by A4, XBOOLE_1:67, XBOOLE_1:81;
then A5: Y c= Z by XBOOLE_1:37;
Y \/ X = X by A2, XBOOLE_1:12;
then A6: Z \ Y c= X by A3, XBOOLE_1:43;
X \ Y misses Z by A1, XBOOLE_1:106;
then Z \ Y = {} by A6, XBOOLE_1:67, XBOOLE_1:81;
then Z c= Y by XBOOLE_1:37;
hence Y = Z by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th6: :: COMBGRAS:6
for X being set
for Y being non empty set
for p being Function of X,Y st p is one-to-one holds
for x1, x2 being Subset of X st x1 <> x2 holds
p .: x1 <> p .: x2
proof
let X be set ; ::_thesis: for Y being non empty set
for p being Function of X,Y st p is one-to-one holds
for x1, x2 being Subset of X st x1 <> x2 holds
p .: x1 <> p .: x2
let Y be non empty set ; ::_thesis: for p being Function of X,Y st p is one-to-one holds
for x1, x2 being Subset of X st x1 <> x2 holds
p .: x1 <> p .: x2
let p be Function of X,Y; ::_thesis: ( p is one-to-one implies for x1, x2 being Subset of X st x1 <> x2 holds
p .: x1 <> p .: x2 )
assume A1: p is one-to-one ; ::_thesis: for x1, x2 being Subset of X st x1 <> x2 holds
p .: x1 <> p .: x2
let x1, x2 be Subset of X; ::_thesis: ( x1 <> x2 implies p .: x1 <> p .: x2 )
A2: X = dom p by FUNCT_2:def_1;
A3: ( not x1 c= x2 implies p .: x1 <> p .: x2 )
proof
assume not x1 c= x2 ; ::_thesis: p .: x1 <> p .: x2
then consider a being set such that
A4: a in x1 and
A5: not a in x2 by TARSKI:def_3;
not p . a in p .: x2
proof
assume p . a in p .: x2 ; ::_thesis: contradiction
then ex b being set st
( b in dom p & b in x2 & p . a = p . b ) by FUNCT_1:def_6;
hence contradiction by A1, A2, A4, A5, FUNCT_1:def_4; ::_thesis: verum
end;
hence p .: x1 <> p .: x2 by A2, A4, FUNCT_1:def_6; ::_thesis: verum
end;
A6: ( not x2 c= x1 implies p .: x1 <> p .: x2 )
proof
assume not x2 c= x1 ; ::_thesis: p .: x1 <> p .: x2
then consider a being set such that
A7: a in x2 and
A8: not a in x1 by TARSKI:def_3;
not p . a in p .: x1
proof
assume p . a in p .: x1 ; ::_thesis: contradiction
then ex b being set st
( b in dom p & b in x1 & p . a = p . b ) by FUNCT_1:def_6;
hence contradiction by A1, A2, A7, A8, FUNCT_1:def_4; ::_thesis: verum
end;
hence p .: x1 <> p .: x2 by A2, A7, FUNCT_1:def_6; ::_thesis: verum
end;
assume x1 <> x2 ; ::_thesis: p .: x1 <> p .: x2
hence p .: x1 <> p .: x2 by A3, A6, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th7: :: COMBGRAS:7
for n being Element of NAT
for a, b, c being set st card a = n - 1 & card b = n - 1 & card c = n - 1 & card (a /\ b) = n - 2 & card (a /\ c) = n - 2 & card (b /\ c) = n - 2 & 2 <= n holds
( ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) & ( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) ) )
proof
let n be Element of NAT ; ::_thesis: for a, b, c being set st card a = n - 1 & card b = n - 1 & card c = n - 1 & card (a /\ b) = n - 2 & card (a /\ c) = n - 2 & card (b /\ c) = n - 2 & 2 <= n holds
( ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) & ( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) ) )
let a, b, c be set ; ::_thesis: ( card a = n - 1 & card b = n - 1 & card c = n - 1 & card (a /\ b) = n - 2 & card (a /\ c) = n - 2 & card (b /\ c) = n - 2 & 2 <= n implies ( ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) & ( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) ) ) )
assume that
A1: card a = n - 1 and
A2: card b = n - 1 and
A3: card c = n - 1 and
A4: card (a /\ b) = n - 2 and
A5: card (a /\ c) = n - 2 and
A6: card (b /\ c) = n - 2 and
A7: 2 <= n ; ::_thesis: ( ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) & ( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) ) )
2 <= n + 1 by A7, NAT_1:13;
then A8: 2 - 1 <= (n + 1) - 1 by XREAL_1:13;
then A9: a is finite by A1, NAT_1:21;
then A10: card (a \ (a /\ b)) = (n - 1) - (n - 2) by A1, A4, CARD_2:44, XBOOLE_1:17;
then consider x1 being set such that
A11: {x1} = a \ (a /\ b) by CARD_2:42;
A12: b is finite by A2, A8, NAT_1:21;
then card (b \ (a /\ b)) = (n - 1) - (n - 2) by A2, A4, CARD_2:44, XBOOLE_1:17;
then consider x2 being set such that
A13: {x2} = b \ (a /\ b) by CARD_2:42;
c is finite by A3, A8, NAT_1:21;
then card (c \ (a /\ c)) = (n - 1) - (n - 2) by A3, A5, CARD_2:44, XBOOLE_1:17;
then consider x3 being set such that
A14: {x3} = c \ (a /\ c) by CARD_2:42;
A15: a = (a /\ b) \/ {x1} by A11, XBOOLE_1:17, XBOOLE_1:45;
A16: (a /\ b) /\ c = (b /\ c) /\ a by XBOOLE_1:16;
A17: a /\ c c= a by XBOOLE_1:17;
A18: (a /\ b) /\ c = (a /\ c) /\ b by XBOOLE_1:16;
A19: b = (a /\ b) \/ {x2} by A13, XBOOLE_1:17, XBOOLE_1:45;
x3 in {x3} by TARSKI:def_1;
then A20: not x3 in a /\ c by A14, XBOOLE_0:def_5;
A21: c = (a /\ c) \/ {x3} by A14, XBOOLE_1:17, XBOOLE_1:45;
A22: x2 in {x2} by TARSKI:def_1;
then A23: not x2 in a /\ b by A13, XBOOLE_0:def_5;
A24: x1 in {x1} by TARSKI:def_1;
then A25: not x1 in a /\ b by A11, XBOOLE_0:def_5;
then A26: x1 <> x2 by A11, A13, A22, XBOOLE_0:def_4;
A27: a /\ b c= b by XBOOLE_1:17;
A28: ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) )
proof
assume 3 <= n ; ::_thesis: ( ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) )
then A29: n - 3 is Element of NAT by NAT_1:21;
A30: ( x1 in c implies ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) )
proof
(a /\ b) /\ c misses {x1}
proof
assume (a /\ b) /\ c meets {x1} ; ::_thesis: contradiction
then not ((a /\ b) /\ c) /\ {x1} = {} by XBOOLE_0:def_7;
then consider x being set such that
A31: x in ((a /\ b) /\ c) /\ {x1} by XBOOLE_0:def_1;
x in {x1} by A31, XBOOLE_0:def_4;
then A32: x = x1 by TARSKI:def_1;
x in (a /\ b) /\ c by A31, XBOOLE_0:def_4;
hence contradiction by A25, A32, XBOOLE_0:def_4; ::_thesis: verum
end;
then A33: (a /\ b) /\ c c= (a /\ c) \ {x1} by A18, XBOOLE_1:17, XBOOLE_1:86;
(a /\ c) \ {x1} c= b
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (a /\ c) \ {x1} or z in b )
assume A34: z in (a /\ c) \ {x1} ; ::_thesis: z in b
then not z in {x1} by XBOOLE_0:def_5;
then ( ( z in a & not z in a \ (a /\ b) & not z in a ) or z in a /\ b ) by A11, A34, XBOOLE_0:def_4, XBOOLE_0:def_5;
hence z in b by XBOOLE_0:def_4; ::_thesis: verum
end;
then (a /\ c) \ {x1} c= (a /\ c) /\ b by XBOOLE_1:19;
then A35: (a /\ c) \ {x1} c= (a /\ b) /\ c by XBOOLE_1:16;
A36: a /\ b misses {x1,x2}
proof
assume a /\ b meets {x1,x2} ; ::_thesis: contradiction
then (a /\ b) /\ {x1,x2} <> {} by XBOOLE_0:def_7;
then consider z1 being set such that
A37: z1 in (a /\ b) /\ {x1,x2} by XBOOLE_0:def_1;
( z1 in a /\ b & z1 in {x1,x2} ) by A37, XBOOLE_0:def_4;
hence contradiction by A25, A23, TARSKI:def_2; ::_thesis: verum
end;
assume x1 in c ; ::_thesis: ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n )
then x1 in a /\ c by A11, A24, XBOOLE_0:def_4;
then A38: {x1} c= a /\ c by ZFMISC_1:31;
a \/ b = (a /\ b) \/ ({x1} \/ {x2}) by A15, A19, XBOOLE_1:5;
then A39: a \/ b = (a /\ b) \/ {x1,x2} by ENUMSET1:1;
card {x1} = 1 by CARD_1:30;
then A40: card ((a /\ c) \ {x1}) = (n - 2) - 1 by A5, A9, A38, CARD_2:44;
then A41: card ((a /\ b) /\ c) = n - 3 by A33, A35, XBOOLE_0:def_10;
x3 = x2
proof
assume A42: x2 <> x3 ; ::_thesis: contradiction
b /\ c c= (a /\ b) /\ c
proof
let z1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not z1 in b /\ c or z1 in (a /\ b) /\ c )
assume A43: z1 in b /\ c ; ::_thesis: z1 in (a /\ b) /\ c
then z1 in b by XBOOLE_0:def_4;
then ( z1 in a /\ b or z1 in {x2} ) by A19, XBOOLE_0:def_3;
then A44: ( z1 in a /\ b or z1 = x2 ) by TARSKI:def_1;
z1 in c by A43, XBOOLE_0:def_4;
then ( z1 in a /\ c or z1 in {x3} ) by A21, XBOOLE_0:def_3;
then ( ( ( z1 in a /\ b or z1 in {x2} ) & z1 in a /\ c ) or ( z1 in a /\ b & ( z1 in a /\ c or z1 in {x3} ) ) ) by A42, A44, TARSKI:def_1;
hence z1 in (a /\ b) /\ c by A27, A13, A14, A18, XBOOLE_0:def_4; ::_thesis: verum
end;
then card (b /\ c) c= card ((a /\ b) /\ c) by CARD_1:11;
then (- 2) + n <= (- 3) + n by A6, A29, A41, NAT_1:39;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then A45: c c= a \/ b by A17, A13, A21, XBOOLE_1:13;
card {x1,x2} = 2 by A26, CARD_2:57;
then card (a \/ b) = (n - 2) + 2 by A4, A9, A39, A36, CARD_2:40;
hence ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) by A40, A33, A35, A45, XBOOLE_0:def_10, XBOOLE_1:12; ::_thesis: verum
end;
( not x1 in c implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) )
proof
A46: x1 <> x3 by A11, A14, A24, A20, XBOOLE_0:def_4;
A47: card (a \ {x1}) = (n - 1) - 1 by A1, A9, A10, A11, CARD_2:44;
assume A48: not x1 in c ; ::_thesis: ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 )
A49: ( a /\ c misses {x1} & a /\ b misses {x1} )
proof
assume ( not a /\ c misses {x1} or not a /\ b misses {x1} ) ; ::_thesis: contradiction
then ( (a /\ c) /\ {x1} <> {} or (a /\ b) /\ {x1} <> {} ) by XBOOLE_0:def_7;
then consider z2 being set such that
A50: ( z2 in (a /\ c) /\ {x1} or z2 in (a /\ b) /\ {x1} ) by XBOOLE_0:def_1;
( ( z2 in a /\ c & z2 in {x1} ) or ( z2 in a /\ b & z2 in {x1} ) ) by A50, XBOOLE_0:def_4;
then ( ( z2 in a & z2 in c & z2 = x1 ) or ( z2 in a /\ b & z2 = x1 ) ) by TARSKI:def_1, XBOOLE_0:def_4;
hence contradiction by A11, A24, A48, XBOOLE_0:def_5; ::_thesis: verum
end;
then a /\ c c= a \ {x1} by XBOOLE_1:17, XBOOLE_1:86;
then A51: a /\ c = a \ {x1} by A5, A9, A47, CARD_FIN:1;
a /\ b c= a \ {x1} by A49, XBOOLE_1:17, XBOOLE_1:86;
then A52: a /\ b = a \ {x1} by A4, A9, A47, CARD_FIN:1;
A53: a /\ b misses {x1,x2,x3}
proof
assume not a /\ b misses {x1,x2,x3} ; ::_thesis: contradiction
then (a /\ b) /\ {x1,x2,x3} <> {} by XBOOLE_0:def_7;
then consider z3 being set such that
A54: z3 in (a /\ b) /\ {x1,x2,x3} by XBOOLE_0:def_1;
( z3 in a /\ b & z3 in {x1,x2,x3} ) by A54, XBOOLE_0:def_4;
hence contradiction by A25, A23, A20, A51, A52, ENUMSET1:def_1; ::_thesis: verum
end;
a \/ b = (a /\ b) \/ ({x1} \/ {x2}) by A15, A19, XBOOLE_1:5;
then a \/ b = (a /\ b) \/ {x1,x2} by ENUMSET1:1;
then (a \/ b) \/ c = (a /\ b) \/ ({x1,x2} \/ {x3}) by A21, A51, A52, XBOOLE_1:5;
then A55: (a \/ b) \/ c = (a /\ b) \/ {x1,x2,x3} by ENUMSET1:3;
(a /\ b) /\ (a /\ c) = a /\ b by A51, A52;
then ((b /\ a) /\ a) /\ c = a /\ b by XBOOLE_1:16;
then A56: (b /\ (a /\ a)) /\ c = a /\ b by XBOOLE_1:16;
then (a /\ b) /\ c = b /\ c by A4, A6, A12, A16, CARD_FIN:1, XBOOLE_1:17;
then x2 <> x3 by A13, A14, A22, A23, A56, XBOOLE_0:def_4;
then card {x1,x2,x3} = 3 by A26, A46, CARD_2:58;
then card ((a \/ b) \/ c) = (n - 2) + 3 by A4, A9, A55, A53, CARD_2:40;
hence ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) by A4, A56; ::_thesis: verum
end;
hence ( ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) by A30; ::_thesis: verum
end;
A57: x1 <> x3 by A11, A14, A24, A20, XBOOLE_0:def_4;
( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) )
proof
assume A58: n = 2 ; ::_thesis: ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 )
then A59: a /\ b = {} by A4;
then (a /\ b) /\ c = a /\ c by A4, A5;
then (a \/ b) \/ c = ((a /\ b) /\ c) \/ ({x1,x2} \/ {x3}) by A11, A13, A14, A59, ENUMSET1:1;
then A60: (a \/ b) \/ c = ((a /\ b) /\ c) \/ {x1,x2,x3} by ENUMSET1:3;
(a /\ b) /\ c = b /\ c by A4, A6, A59;
then x2 <> x3 by A13, A14, A22, A59, XBOOLE_0:def_4;
hence ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) by A26, A57, A58, A59, A60, CARD_2:58; ::_thesis: verum
end;
hence ( ( not 3 <= n or ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) or ( card ((a /\ b) /\ c) = n - 3 & card ((a \/ b) \/ c) = n ) ) & ( n = 2 implies ( card ((a /\ b) /\ c) = n - 2 & card ((a \/ b) \/ c) = n + 1 ) ) ) by A28; ::_thesis: verum
end;
theorem Th8: :: COMBGRAS:8
for P1, P2 being IncProjStr st IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) holds
for A1 being POINT of P1
for A2 being POINT of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
proof
let P1, P2 be IncProjStr ; ::_thesis: ( IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) implies for A1 being POINT of P1
for A2 being POINT of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2 )
assume A1: IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) ; ::_thesis: for A1 being POINT of P1
for A2 being POINT of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let A1 be POINT of P1; ::_thesis: for A2 being POINT of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let A2 be POINT of P2; ::_thesis: ( A1 = A2 implies for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2 )
assume A2: A1 = A2 ; ::_thesis: for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let L1 be LINE of P1; ::_thesis: for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let L2 be LINE of P2; ::_thesis: ( L1 = L2 & A1 on L1 implies A2 on L2 )
assume ( L1 = L2 & A1 on L1 ) ; ::_thesis: A2 on L2
then [A2,L2] in the Inc of P2 by A1, A2, INCSP_1:def_1;
hence A2 on L2 by INCSP_1:def_1; ::_thesis: verum
end;
theorem Th9: :: COMBGRAS:9
for P1, P2 being IncProjStr st IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) holds
for A1 being Subset of the Points of P1
for A2 being Subset of the Points of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
proof
let P1, P2 be IncProjStr ; ::_thesis: ( IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) implies for A1 being Subset of the Points of P1
for A2 being Subset of the Points of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2 )
assume A1: IncProjStr(# the Points of P1, the Lines of P1, the Inc of P1 #) = IncProjStr(# the Points of P2, the Lines of P2, the Inc of P2 #) ; ::_thesis: for A1 being Subset of the Points of P1
for A2 being Subset of the Points of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let A1 be Subset of the Points of P1; ::_thesis: for A2 being Subset of the Points of P2 st A1 = A2 holds
for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let A2 be Subset of the Points of P2; ::_thesis: ( A1 = A2 implies for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2 )
assume A2: A1 = A2 ; ::_thesis: for L1 being LINE of P1
for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let L1 be LINE of P1; ::_thesis: for L2 being LINE of P2 st L1 = L2 & A1 on L1 holds
A2 on L2
let L2 be LINE of P2; ::_thesis: ( L1 = L2 & A1 on L1 implies A2 on L2 )
assume that
A3: L1 = L2 and
A4: A1 on L1 ; ::_thesis: A2 on L2
thus A2 on L2 ::_thesis: verum
proof
let A be POINT of P2; :: according to INCSP_1:def_4 ::_thesis: ( not A in A2 or A on L2 )
consider B being POINT of P1 such that
A5: A = B by A1;
assume A in A2 ; ::_thesis: A on L2
then B on L1 by A2, A4, A5, INCSP_1:def_4;
then [A,L2] in the Inc of P2 by A1, A3, A5, INCSP_1:def_1;
hence A on L2 by INCSP_1:def_1; ::_thesis: verum
end;
end;
registration
cluster strict with_non-trivial_lines linear up-2-rank for IncProjStr ;
existence
ex b1 being IncProjStr st
( b1 is with_non-trivial_lines & b1 is linear & b1 is up-2-rank & b1 is strict )
proof
set P = the strict IncSpace-like IncStruct ;
take IT = IncProjStr(# the Points of the strict IncSpace-like IncStruct , the Lines of the strict IncSpace-like IncStruct , the Inc of the strict IncSpace-like IncStruct #); ::_thesis: ( IT is with_non-trivial_lines & IT is linear & IT is up-2-rank & IT is strict )
thus for L being LINE of IT ex A, B being POINT of IT st
( A <> B & {A,B} on L ) :: according to INCSP_1:def_8 ::_thesis: ( IT is linear & IT is up-2-rank & IT is strict )
proof
let L be LINE of IT; ::_thesis: ex A, B being POINT of IT st
( A <> B & {A,B} on L )
reconsider L9 = L as LINE of the strict IncSpace-like IncStruct ;
consider A9, B9 being POINT of the strict IncSpace-like IncStruct such that
A1: ( A9 <> B9 & {A9,B9} on L9 ) by INCSP_1:def_8;
reconsider A = A9, B = B9 as POINT of IT ;
take A ; ::_thesis: ex B being POINT of IT st
( A <> B & {A,B} on L )
take B ; ::_thesis: ( A <> B & {A,B} on L )
thus ( A <> B & {A,B} on L ) by A1, Th9; ::_thesis: verum
end;
thus IT is linear ::_thesis: ( IT is up-2-rank & IT is strict )
proof
let A, B be POINT of IT; :: according to INCPROJ:def_5 ::_thesis: ex b1 being Element of the Lines of IT st
( A on b1 & B on b1 )
reconsider A9 = A, B9 = B as POINT of the strict IncSpace-like IncStruct ;
consider L9 being LINE of the strict IncSpace-like IncStruct such that
A2: {A9,B9} on L9 by INCSP_1:def_9;
reconsider L = L9 as LINE of IT ;
take L ; ::_thesis: ( A on L & B on L )
( A9 on L9 & B9 on L9 ) by A2, INCSP_1:1;
hence ( A on L & B on L ) by Th8; ::_thesis: verum
end;
thus for A, B being POINT of IT
for K, L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds
K = L :: according to INCSP_1:def_10 ::_thesis: IT is strict
proof
let A, B be POINT of IT; ::_thesis: for K, L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds
K = L
let K, L be LINE of IT; ::_thesis: ( A <> B & {A,B} on K & {A,B} on L implies K = L )
assume that
A3: A <> B and
A4: ( {A,B} on K & {A,B} on L ) ; ::_thesis: K = L
reconsider K9 = K, L9 = L as LINE of the strict IncSpace-like IncStruct ;
reconsider A9 = A, B9 = B as POINT of the strict IncSpace-like IncStruct ;
( {A9,B9} on K9 & {A9,B9} on L9 ) by A4, Th9;
hence K = L by A3, INCSP_1:def_10; ::_thesis: verum
end;
thus IT is strict ; ::_thesis: verum
end;
end;
begin
definition
mode PartialLinearSpace is with_non-trivial_lines up-2-rank IncProjStr ;
end;
definition
let k be Element of NAT ;
let X be non empty set ;
assume that
B1: 0 < k and
B2: k + 1 c= card X ;
func G_ (k,X) -> strict PartialLinearSpace means :Def1: :: COMBGRAS:def 1
( the Points of it = { A where A is Subset of X : card A = k } & the Lines of it = { L where L is Subset of X : card L = k + 1 } & the Inc of it = (RelIncl (bool X)) /\ [: the Points of it, the Lines of it:] );
existence
ex b1 being strict PartialLinearSpace st
( the Points of b1 = { A where A is Subset of X : card A = k } & the Lines of b1 = { L where L is Subset of X : card L = k + 1 } & the Inc of b1 = (RelIncl (bool X)) /\ [: the Points of b1, the Lines of b1:] )
proof
set L = { B where B is Subset of X : card B = k + 1 } ;
set P = { A where A is Subset of X : card A = k } ;
set I = (RelIncl (bool X)) /\ [: { A where A is Subset of X : card A = k } , { B where B is Subset of X : card B = k + 1 } :];
consider B being set such that
A1: ( B c= X & card B = k + 1 ) by B2, CARD_FIL:36;
B in { B where B is Subset of X : card B = k + 1 } by A1;
then reconsider L = { B where B is Subset of X : card B = k + 1 } as non empty set ;
k <= k + 1 by NAT_1:11;
then k c= k + 1 by NAT_1:39;
then k c= card X by B2, XBOOLE_1:1;
then consider A being set such that
A2: ( A c= X & card A = k ) by CARD_FIL:36;
A in { A where A is Subset of X : card A = k } by A2;
then reconsider P = { A where A is Subset of X : card A = k } as non empty set ;
reconsider I = (RelIncl (bool X)) /\ [: { A where A is Subset of X : card A = k } , { B where B is Subset of X : card B = k + 1 } :] as Relation of P,L by XBOOLE_1:17;
set G = IncProjStr(# P,L,I #);
A3: IncProjStr(# P,L,I #) is up-2-rank
proof
let a, b be POINT of IncProjStr(# P,L,I #); :: according to INCSP_1:def_10 ::_thesis: for b1, b2 being Element of the Lines of IncProjStr(# P,L,I #) holds
( a = b or not {a,b} on b1 or not {a,b} on b2 or b1 = b2 )
let l1, l2 be LINE of IncProjStr(# P,L,I #); ::_thesis: ( a = b or not {a,b} on l1 or not {a,b} on l2 or l1 = l2 )
assume that
A4: a <> b and
A5: {a,b} on l1 and
A6: {a,b} on l2 ; ::_thesis: l1 = l2
b in P ;
then A7: ex B being Subset of X st
( b = B & card B = k ) ;
a in P ;
then A8: ex A being Subset of X st
( a = A & card A = k ) ;
then A9: k + 1 c= card (a \/ b) by A7, A4, Th1;
l1 in L ;
then A10: ex C being Subset of X st
( l1 = C & card C = k + 1 ) ;
then A11: l1 is finite ;
A12: b in {a,b} by TARSKI:def_2;
then b on l1 by A5, INCSP_1:def_4;
then [b,l1] in I by INCSP_1:def_1;
then [b,l1] in RelIncl (bool X) by XBOOLE_0:def_4;
then A13: b c= l1 by A7, A10, WELLORD2:def_1;
l2 in L ;
then A14: ex D being Subset of X st
( l2 = D & card D = k + 1 ) ;
then A15: l2 is finite ;
A16: a in {a,b} by TARSKI:def_2;
then a on l2 by A6, INCSP_1:def_4;
then [a,l2] in I by INCSP_1:def_1;
then [a,l2] in RelIncl (bool X) by XBOOLE_0:def_4;
then A17: a c= l2 by A8, A14, WELLORD2:def_1;
b on l2 by A12, A6, INCSP_1:def_4;
then [b,l2] in I by INCSP_1:def_1;
then [b,l2] in RelIncl (bool X) by XBOOLE_0:def_4;
then A18: b c= l2 by A7, A14, WELLORD2:def_1;
then a \/ b c= l2 by A17, XBOOLE_1:8;
then card (a \/ b) c= k + 1 by A14, CARD_1:11;
then A19: card (a \/ b) = k + 1 by A9, XBOOLE_0:def_10;
a on l1 by A16, A5, INCSP_1:def_4;
then [a,l1] in I by INCSP_1:def_1;
then [a,l1] in RelIncl (bool X) by XBOOLE_0:def_4;
then a c= l1 by A8, A10, WELLORD2:def_1;
then a \/ b = l1 by A10, A13, A19, A11, CARD_FIN:1, XBOOLE_1:8;
hence l1 = l2 by A14, A17, A18, A19, A15, CARD_FIN:1, XBOOLE_1:8; ::_thesis: verum
end;
IncProjStr(# P,L,I #) is with_non-trivial_lines
proof
let l be LINE of IncProjStr(# P,L,I #); :: according to INCSP_1:def_8 ::_thesis: ex b1, b2 being Element of the Points of IncProjStr(# P,L,I #) st
( not b1 = b2 & {b1,b2} on l )
l in L ;
then consider C being Subset of X such that
A20: l = C and
A21: card C = k + 1 ;
1 < k + 1 by B1, XREAL_1:29;
then 1 + 1 <= k + 1 by NAT_1:13;
then 2 c= k + 1 by NAT_1:39;
then consider a, b being set such that
A22: a in C and
A23: b in C and
A24: a <> b by A21, PENCIL_1:2;
reconsider x = C \ {a}, y = C \ {b} as Subset of X ;
card x = k by A21, A22, STIRL2_1:55;
then A25: x in P ;
card y = k by A21, A23, STIRL2_1:55;
then y in P ;
then reconsider x = x, y = y as POINT of IncProjStr(# P,L,I #) by A25;
take x ; ::_thesis: ex b1 being Element of the Points of IncProjStr(# P,L,I #) st
( not x = b1 & {x,b1} on l )
take y ; ::_thesis: ( not x = y & {x,y} on l )
not b in {a} by A24, TARSKI:def_1;
then ( b in {b} & b in x ) by A23, TARSKI:def_1, XBOOLE_0:def_5;
hence x <> y by XBOOLE_0:def_5; ::_thesis: {x,y} on l
A26: ( C c= {a} \/ C & C c= {b} \/ C ) by XBOOLE_1:7;
{x,y} on l
proof
let z be POINT of IncProjStr(# P,L,I #); :: according to INCSP_1:def_4 ::_thesis: ( not z in {x,y} or z on l )
assume z in {x,y} ; ::_thesis: z on l
then A27: ( z = x or z = y ) by TARSKI:def_2;
then z c= l by A20, A26, XBOOLE_1:43;
then [z,l] in RelIncl (bool X) by A20, A27, WELLORD2:def_1;
then [z,l] in I by XBOOLE_0:def_4;
hence z on l by INCSP_1:def_1; ::_thesis: verum
end;
hence {x,y} on l ; ::_thesis: verum
end;
hence ex b1 being strict PartialLinearSpace st
( the Points of b1 = { A where A is Subset of X : card A = k } & the Lines of b1 = { L where L is Subset of X : card L = k + 1 } & the Inc of b1 = (RelIncl (bool X)) /\ [: the Points of b1, the Lines of b1:] ) by A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict PartialLinearSpace st the Points of b1 = { A where A is Subset of X : card A = k } & the Lines of b1 = { L where L is Subset of X : card L = k + 1 } & the Inc of b1 = (RelIncl (bool X)) /\ [: the Points of b1, the Lines of b1:] & the Points of b2 = { A where A is Subset of X : card A = k } & the Lines of b2 = { L where L is Subset of X : card L = k + 1 } & the Inc of b2 = (RelIncl (bool X)) /\ [: the Points of b2, the Lines of b2:] holds
b1 = b2 ;
end;
:: deftheorem Def1 defines G_ COMBGRAS:def_1_:_
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for b3 being strict PartialLinearSpace holds
( b3 = G_ (k,X) iff ( the Points of b3 = { A where A is Subset of X : card A = k } & the Lines of b3 = { L where L is Subset of X : card L = k + 1 } & the Inc of b3 = (RelIncl (bool X)) /\ [: the Points of b3, the Lines of b3:] ) );
theorem Th10: :: COMBGRAS:10
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for A being POINT of (G_ (k,X))
for L being LINE of (G_ (k,X)) holds
( A on L iff A c= L )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for A being POINT of (G_ (k,X))
for L being LINE of (G_ (k,X)) holds
( A on L iff A c= L )
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for A being POINT of (G_ (k,X))
for L being LINE of (G_ (k,X)) holds
( A on L iff A c= L ) )
assume A1: ( 0 < k & k + 1 c= card X ) ; ::_thesis: for A being POINT of (G_ (k,X))
for L being LINE of (G_ (k,X)) holds
( A on L iff A c= L )
then A2: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by Def1;
let A be POINT of (G_ (k,X)); ::_thesis: for L being LINE of (G_ (k,X)) holds
( A on L iff A c= L )
A in the Points of (G_ (k,X)) ;
then A3: ex A1 being Subset of X st
( A1 = A & card A1 = k ) by A2;
A4: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, Def1;
let L be LINE of (G_ (k,X)); ::_thesis: ( A on L iff A c= L )
L in the Lines of (G_ (k,X)) ;
then A5: ex L1 being Subset of X st
( L1 = L & card L1 = k + 1 ) by A4;
A6: the Inc of (G_ (k,X)) = (RelIncl (bool X)) /\ [: the Points of (G_ (k,X)), the Lines of (G_ (k,X)):] by A1, Def1;
thus ( A on L implies A c= L ) ::_thesis: ( A c= L implies A on L )
proof
assume A on L ; ::_thesis: A c= L
then [A,L] in the Inc of (G_ (k,X)) by INCSP_1:def_1;
then [A,L] in RelIncl (bool X) by A6, XBOOLE_0:def_4;
hence A c= L by A3, A5, WELLORD2:def_1; ::_thesis: verum
end;
thus ( A c= L implies A on L ) ::_thesis: verum
proof
assume A c= L ; ::_thesis: A on L
then [A,L] in RelIncl (bool X) by A3, A5, WELLORD2:def_1;
then [A,L] in the Inc of (G_ (k,X)) by A6, XBOOLE_0:def_4;
hence A on L by INCSP_1:def_1; ::_thesis: verum
end;
end;
theorem Th11: :: COMBGRAS:11
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
G_ (k,X) is Vebleian
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
G_ (k,X) is Vebleian
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies G_ (k,X) is Vebleian )
k <= k + 1 by NAT_1:11;
then A1: k c= k + 1 by NAT_1:39;
assume A2: ( 0 < k & k + 1 c= card X ) ; ::_thesis: G_ (k,X) is Vebleian
then A3: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by Def1;
let A1, A2, A3, A4, A5, A6 be POINT of (G_ (k,X)); :: according to INCPROJ:def_8 ::_thesis: for b1, b2, b3, b4 being Element of the Lines of (G_ (k,X)) holds
( not A1 on b1 or not A2 on b1 or not A3 on b2 or not A4 on b2 or not A5 on b1 or not A5 on b2 or not A1 on b3 or not A3 on b3 or not A2 on b4 or not A4 on b4 or A5 on b3 or A5 on b4 or b1 = b2 or ex b5 being Element of the Points of (G_ (k,X)) st
( b5 on b3 & b5 on b4 ) )
let L1, L2, L3, L4 be LINE of (G_ (k,X)); ::_thesis: ( not A1 on L1 or not A2 on L1 or not A3 on L2 or not A4 on L2 or not A5 on L1 or not A5 on L2 or not A1 on L3 or not A3 on L3 or not A2 on L4 or not A4 on L4 or A5 on L3 or A5 on L4 or L1 = L2 or ex b1 being Element of the Points of (G_ (k,X)) st
( b1 on L3 & b1 on L4 ) )
assume that
A4: A1 on L1 and
A5: A2 on L1 and
A6: A3 on L2 and
A7: A4 on L2 and
A8: ( A5 on L1 & A5 on L2 ) and
A9: A1 on L3 and
A10: A3 on L3 and
A11: A2 on L4 and
A12: A4 on L4 and
A13: ( not A5 on L3 & not A5 on L4 ) and
A14: L1 <> L2 ; ::_thesis: ex b1 being Element of the Points of (G_ (k,X)) st
( b1 on L3 & b1 on L4 )
A15: ( A2 c= L1 & A4 c= L2 ) by A2, A5, A7, Th10;
A16: ( A1 <> A3 & A2 <> A4 )
proof
assume ( A1 = A3 or A2 = A4 ) ; ::_thesis: contradiction
then ( ( {A1,A5} on L1 & {A1,A5} on L2 ) or ( {A2,A5} on L1 & {A2,A5} on L2 ) ) by A4, A5, A6, A7, A8, INCSP_1:1;
hence contradiction by A9, A11, A13, A14, INCSP_1:def_10; ::_thesis: verum
end;
A17: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A2, Def1;
( A5 c= L1 & A5 c= L2 ) by A2, A8, Th10;
then A18: A5 c= L1 /\ L2 by XBOOLE_1:19;
A5 in the Points of (G_ (k,X)) ;
then ex B5 being Subset of X st
( B5 = A5 & card B5 = k ) by A3;
then A19: k c= card (L1 /\ L2) by A18, CARD_1:11;
L2 in the Lines of (G_ (k,X)) ;
then A20: ex K2 being Subset of X st
( K2 = L2 & card K2 = k + 1 ) by A17;
A3 in the Points of (G_ (k,X)) ;
then A21: ex B3 being Subset of X st
( B3 = A3 & card B3 = k ) by A3;
A1 in the Points of (G_ (k,X)) ;
then ex B1 being Subset of X st
( B1 = A1 & card B1 = k ) by A3;
then A22: k + 1 c= card (A1 \/ A3) by A21, A16, Th1;
A23: ( A1 c= L1 & A3 c= L2 ) by A2, A4, A6, Th10;
L3 in the Lines of (G_ (k,X)) ;
then A24: ex K3 being Subset of X st
( K3 = L3 & card K3 = k + 1 ) by A17;
then A25: L3 is finite ;
A4 in the Points of (G_ (k,X)) ;
then A26: ex B4 being Subset of X st
( B4 = A4 & card B4 = k ) by A3;
A2 in the Points of (G_ (k,X)) ;
then ex B2 being Subset of X st
( B2 = A2 & card B2 = k ) by A3;
then A27: k + 1 c= card (A2 \/ A4) by A26, A16, Th1;
L4 in the Lines of (G_ (k,X)) ;
then A28: ex K4 being Subset of X st
( K4 = L4 & card K4 = k + 1 ) by A17;
then A29: L4 is finite ;
A30: ( A2 c= L4 & A4 c= L4 ) by A2, A11, A12, Th10;
then A2 \/ A4 c= L4 by XBOOLE_1:8;
then card (A2 \/ A4) c= k + 1 by A28, CARD_1:11;
then card (A2 \/ A4) = k + 1 by A27, XBOOLE_0:def_10;
then A2 \/ A4 = L4 by A28, A30, A29, CARD_FIN:1, XBOOLE_1:8;
then A31: L4 c= L1 \/ L2 by A15, XBOOLE_1:13;
L1 in the Lines of (G_ (k,X)) ;
then A32: ex K1 being Subset of X st
( K1 = L1 & card K1 = k + 1 ) by A17;
then card (L1 /\ L2) in k + 1 by A20, A14, Th1;
then card (L1 /\ L2) in succ k by NAT_1:38;
then card (L1 /\ L2) c= k by ORDINAL1:22;
then card (L1 /\ L2) = k by A19, XBOOLE_0:def_10;
then A33: card (L1 \/ L2) = k + (2 * 1) by A32, A20, Th2;
A34: ( A1 c= L3 & A3 c= L3 ) by A2, A9, A10, Th10;
then A1 \/ A3 c= L3 by XBOOLE_1:8;
then card (A1 \/ A3) c= k + 1 by A24, CARD_1:11;
then card (A1 \/ A3) = k + 1 by A22, XBOOLE_0:def_10;
then A1 \/ A3 = L3 by A24, A34, A25, CARD_FIN:1, XBOOLE_1:8;
then L3 c= L1 \/ L2 by A23, XBOOLE_1:13;
then L3 \/ L4 c= L1 \/ L2 by A31, XBOOLE_1:8;
then card (L3 \/ L4) c= k + 2 by A33, CARD_1:11;
then card (L3 \/ L4) in succ (k + 2) by ORDINAL1:22;
then ( card (L3 \/ L4) in (k + 1) + 1 or card (L3 \/ L4) = k + 2 ) by ORDINAL1:8;
then ( card (L3 \/ L4) in succ (k + 1) or card (L3 \/ L4) = k + 2 ) by NAT_1:38;
then A35: ( card (L3 \/ L4) c= k + 1 or card (L3 \/ L4) = k + 2 ) by ORDINAL1:22;
k + 1 c= card (L3 \/ L4) by A24, CARD_1:11, XBOOLE_1:7;
then ( card (L3 \/ L4) = (k + 1) + (2 * 0) or card (L3 \/ L4) = k + (2 * 1) ) by A35, XBOOLE_0:def_10;
then k c= card (L3 /\ L4) by A24, A28, A1, Th2;
then consider B6 being set such that
A36: B6 c= L3 /\ L4 and
A37: card B6 = k by CARD_FIL:36;
A38: L3 /\ L4 c= L3 by XBOOLE_1:17;
then L3 /\ L4 c= X by A24, XBOOLE_1:1;
then reconsider A6 = B6 as Subset of X by A36, XBOOLE_1:1;
A39: A6 in the Points of (G_ (k,X)) by A3, A37;
L3 /\ L4 c= L4 by XBOOLE_1:17;
then A40: B6 c= L4 by A36, XBOOLE_1:1;
reconsider A6 = A6 as POINT of (G_ (k,X)) by A39;
take B6 ; ::_thesis: ( B6 is Element of the Points of (G_ (k,X)) & B6 on L3 & B6 on L4 )
( A6 c= B6 & B6 c= L3 ) by A36, A38, XBOOLE_1:1;
hence ( B6 is Element of the Points of (G_ (k,X)) & B6 on L3 & B6 on L4 ) by A2, A40, Th10; ::_thesis: verum
end;
theorem Th12: :: COMBGRAS:12
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for A1, A2, A3, A4, A5, A6 being POINT of (G_ (k,X))
for L1, L2, L3, L4 being LINE of (G_ (k,X)) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 holds
ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for A1, A2, A3, A4, A5, A6 being POINT of (G_ (k,X))
for L1, L2, L3, L4 being LINE of (G_ (k,X)) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 holds
ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) )
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for A1, A2, A3, A4, A5, A6 being POINT of (G_ (k,X))
for L1, L2, L3, L4 being LINE of (G_ (k,X)) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 holds
ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) ) )
assume that
A1: 0 < k and
A2: k + 1 c= card X ; ::_thesis: for A1, A2, A3, A4, A5, A6 being POINT of (G_ (k,X))
for L1, L2, L3, L4 being LINE of (G_ (k,X)) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 holds
ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) )
A3: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, A2, Def1;
A4: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A2, Def1;
let A1, A2, A3, A4, A5, A6 be POINT of (G_ (k,X)); ::_thesis: for L1, L2, L3, L4 being LINE of (G_ (k,X)) st A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 holds
ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) )
let L1, L2, L3, L4 be LINE of (G_ (k,X)); ::_thesis: ( A1 on L1 & A2 on L1 & A3 on L2 & A4 on L2 & A5 on L1 & A5 on L2 & A1 on L3 & A3 on L3 & A2 on L4 & A4 on L4 & not A5 on L3 & not A5 on L4 & L1 <> L2 & L3 <> L4 implies ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) ) )
assume that
A5: A1 on L1 and
A6: A2 on L1 and
A7: A3 on L2 and
A8: A4 on L2 and
A9: A5 on L1 and
A10: A5 on L2 and
A11: A1 on L3 and
A12: A3 on L3 and
A13: A2 on L4 and
A14: A4 on L4 and
A15: not A5 on L3 and
A16: not A5 on L4 and
A17: L1 <> L2 and
A18: L3 <> L4 ; ::_thesis: ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) )
A19: ( A1 c= L1 & A2 c= L1 ) by A1, A2, A5, A6, Th10;
A20: ( A3 c= L2 & A4 c= L2 ) by A1, A2, A7, A8, Th10;
A21: ( A5 c= L1 & A5 c= L2 ) by A1, A2, A9, A10, Th10;
A5 in the Points of (G_ (k,X)) ;
then A22: ex B5 being Subset of X st
( B5 = A5 & card B5 = k ) by A3;
A2 in the Points of (G_ (k,X)) ;
then A23: ex B2 being Subset of X st
( B2 = A2 & card B2 = k ) by A3;
then A24: A2 is finite ;
A25: k - 1 is Element of NAT by A1, NAT_1:20;
L3 in the Lines of (G_ (k,X)) ;
then A26: ex K3 being Subset of X st
( K3 = L3 & card K3 = k + 1 ) by A4;
then A27: L3 is finite ;
L4 in the Lines of (G_ (k,X)) ;
then ex K4 being Subset of X st
( K4 = L4 & card K4 = k + 1 ) by A4;
then card (L3 /\ L4) in k + 1 by A26, A18, Th1;
then card (L3 /\ L4) in succ k by NAT_1:38;
then A28: card (L3 /\ L4) c= k by ORDINAL1:22;
A1 in the Points of (G_ (k,X)) ;
then A29: ex B1 being Subset of X st
( B1 = A1 & card B1 = k ) by A3;
then A30: A1 is finite ;
G_ (k,X) is Vebleian by A1, A2, Th11;
then consider A6 being POINT of (G_ (k,X)) such that
A31: A6 on L3 and
A32: A6 on L4 by A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, INCPROJ:def_8;
A6 in the Points of (G_ (k,X)) ;
then A33: ex a6 being Subset of X st
( a6 = A6 & card a6 = k ) by A3;
then A34: A6 is finite ;
A35: ( A6 c= L3 & A6 c= L4 ) by A1, A2, A31, A32, Th10;
then A6 c= L3 /\ L4 by XBOOLE_1:19;
then k c= card (L3 /\ L4) by A33, CARD_1:11;
then card (L3 /\ L4) = k by A28, XBOOLE_0:def_10;
then A36: L3 /\ L4 = A6 by A33, A35, A27, CARD_FIN:1, XBOOLE_1:19;
L2 in the Lines of (G_ (k,X)) ;
then A37: ex K2 being Subset of X st
( K2 = L2 & card K2 = k + 1 ) by A4;
then A38: L2 is finite ;
A4 in the Points of (G_ (k,X)) ;
then A39: ex B4 being Subset of X st
( B4 = A4 & card B4 = k ) by A3;
then A40: A4 is finite ;
L1 in the Lines of (G_ (k,X)) ;
then A41: ex K1 being Subset of X st
( K1 = L1 & card K1 = k + 1 ) by A4;
then A42: L1 is finite ;
A3 in the Points of (G_ (k,X)) ;
then A43: ex B3 being Subset of X st
( B3 = A3 & card B3 = k ) by A3;
then A44: A3 is finite ;
A45: ( A3 c= L3 & A4 c= L4 ) by A1, A2, A12, A14, Th10;
then A46: A3 /\ A4 c= A6 by A36, XBOOLE_1:27;
A47: ( A1 c= L3 & A2 c= L4 ) by A1, A2, A11, A13, Th10;
then A48: A1 /\ A2 c= A6 by A36, XBOOLE_1:27;
then A49: (A1 /\ A2) \/ (A3 /\ A4) c= A6 by A46, XBOOLE_1:8;
A50: ( not A6 on L1 implies A6 = (A1 /\ A2) \/ (A3 /\ A4) )
proof
assume A51: not A6 on L1 ; ::_thesis: A6 = (A1 /\ A2) \/ (A3 /\ A4)
A52: ( not A6 on L2 implies A6 = (A1 /\ A2) \/ (A3 /\ A4) )
proof
A53: A1 \/ A2 c= L1 by A19, XBOOLE_1:8;
then A54: card (A1 \/ A2) c= k + 1 by A41, CARD_1:11;
A55: A3 \/ A4 c= L2 by A20, XBOOLE_1:8;
then A56: card (A3 \/ A4) c= k + 1 by A37, CARD_1:11;
A57: card A3 = (k - 1) + 1 by A43;
card ((A1 /\ A2) \/ (A3 /\ A4)) c= k by A33, A49, CARD_1:11;
then card ((A1 /\ A2) \/ (A3 /\ A4)) in succ k by ORDINAL1:22;
then ( card ((A1 /\ A2) \/ (A3 /\ A4)) in k or card ((A1 /\ A2) \/ (A3 /\ A4)) = k ) by ORDINAL1:8;
then ( card ((A1 /\ A2) \/ (A3 /\ A4)) in succ (k - 1) or card ((A1 /\ A2) \/ (A3 /\ A4)) = k ) by A25, A57, NAT_1:38;
then A58: ( card ((A1 /\ A2) \/ (A3 /\ A4)) c= k - 1 or card ((A1 /\ A2) \/ (A3 /\ A4)) = k ) by A25, ORDINAL1:22;
A59: card A1 = (k - 1) + 1 by A29;
assume A60: not A6 on L2 ; ::_thesis: A6 = (A1 /\ A2) \/ (A3 /\ A4)
A61: ( A1 <> A2 & A3 <> A4 )
proof
assume ( A1 = A2 or A3 = A4 ) ; ::_thesis: contradiction
then ( ( {A1,A6} on L3 & {A1,A6} on L4 ) or ( {A3,A6} on L3 & {A3,A6} on L4 ) ) by A11, A12, A13, A14, A31, A32, INCSP_1:1;
hence contradiction by A5, A7, A18, A51, A60, INCSP_1:def_10; ::_thesis: verum
end;
then k + 1 c= card (A1 \/ A2) by A29, A23, Th1;
then card (A1 \/ A2) = (k - 1) + (2 * 1) by A54, XBOOLE_0:def_10;
then A62: card (A1 /\ A2) = k - 1 by A23, A25, A59, Th2;
k + 1 c= card (A3 \/ A4) by A43, A39, A61, Th1;
then card (A3 \/ A4) = (k - 1) + (2 * 1) by A56, XBOOLE_0:def_10;
then A63: card (A3 /\ A4) = k - 1 by A39, A25, A57, Th2;
A64: not card ((A1 /\ A2) \/ (A3 /\ A4)) = k - 1
proof
A65: A5 c= L1 /\ L2 by A21, XBOOLE_1:19;
A66: (A1 /\ A2) /\ (A3 /\ A4) c= A1 /\ A2 by XBOOLE_1:17;
A67: ((A1 /\ A2) /\ A3) /\ A4 = (A1 /\ A2) /\ (A3 /\ A4) by XBOOLE_1:16;
A68: A1 /\ A2 c= A1 by XBOOLE_1:17;
then A69: A1 = (((A1 /\ A2) /\ A3) /\ A4) \/ (A1 \ (((A1 /\ A2) /\ A3) /\ A4)) by A66, A67, XBOOLE_1:1, XBOOLE_1:45;
assume A70: card ((A1 /\ A2) \/ (A3 /\ A4)) = k - 1 ; ::_thesis: contradiction
then card ((A1 /\ A2) \/ (A3 /\ A4)) = (k - 1) + (2 * 0) ;
then A71: card ((A1 /\ A2) /\ (A3 /\ A4)) = k - 1 by A25, A62, A63, Th2;
then A72: (A1 /\ A2) /\ (A3 /\ A4) = (A1 /\ A2) \/ (A3 /\ A4) by A30, A44, A70, CARD_FIN:1, XBOOLE_1:29;
then card (A1 \ (((A1 /\ A2) /\ A3) /\ A4)) = k - (k - 1) by A29, A30, A70, A68, A66, A67, CARD_2:44, XBOOLE_1:1;
then consider x1 being set such that
A73: A1 \ (((A1 /\ A2) /\ A3) /\ A4) = {x1} by CARD_2:42;
A74: A1 /\ A2 c= A2 by XBOOLE_1:17;
then A75: A2 = (((A1 /\ A2) /\ A3) /\ A4) \/ (A2 \ (((A1 /\ A2) /\ A3) /\ A4)) by A66, A67, XBOOLE_1:1, XBOOLE_1:45;
card (A2 \ (((A1 /\ A2) /\ A3) /\ A4)) = k - (k - 1) by A23, A24, A70, A74, A66, A72, A67, CARD_2:44, XBOOLE_1:1;
then consider x2 being set such that
A76: A2 \ (((A1 /\ A2) /\ A3) /\ A4) = {x2} by CARD_2:42;
x1 in {x1} by TARSKI:def_1;
then A77: not x1 in ((A1 /\ A2) /\ A3) /\ A4 by A73, XBOOLE_0:def_5;
A78: (A1 /\ A2) /\ (A3 /\ A4) c= A3 /\ A4 by XBOOLE_1:17;
A79: A3 /\ A4 c= A4 by XBOOLE_1:17;
then A80: A4 = (((A1 /\ A2) /\ A3) /\ A4) \/ (A4 \ (((A1 /\ A2) /\ A3) /\ A4)) by A78, A67, XBOOLE_1:1, XBOOLE_1:45;
card (A4 \ (((A1 /\ A2) /\ A3) /\ A4)) = k - (k - 1) by A39, A40, A70, A79, A78, A72, A67, CARD_2:44, XBOOLE_1:1;
then consider x4 being set such that
A81: A4 \ (((A1 /\ A2) /\ A3) /\ A4) = {x4} by CARD_2:42;
A82: A3 /\ A4 c= A3 by XBOOLE_1:17;
then A83: A3 = (((A1 /\ A2) /\ A3) /\ A4) \/ (A3 \ (((A1 /\ A2) /\ A3) /\ A4)) by A78, A67, XBOOLE_1:1, XBOOLE_1:45;
card (A3 \ (((A1 /\ A2) /\ A3) /\ A4)) = k - (k - 1) by A43, A44, A70, A82, A78, A72, A67, CARD_2:44, XBOOLE_1:1;
then consider x3 being set such that
A84: A3 \ (((A1 /\ A2) /\ A3) /\ A4) = {x3} by CARD_2:42;
( k + 1 c= card (A3 \/ A4) & card (A3 \/ A4) c= k + 1 ) by A43, A39, A37, A61, A55, Th1, CARD_1:11;
then card (A3 \/ A4) = k + 1 by XBOOLE_0:def_10;
then A3 \/ A4 = L2 by A37, A20, A38, CARD_FIN:1, XBOOLE_1:8;
then A85: L2 = (((A1 /\ A2) /\ A3) /\ A4) \/ ({x3} \/ {x4}) by A84, A81, A83, A80, XBOOLE_1:5;
then A86: L2 = (((A1 /\ A2) /\ A3) /\ A4) \/ {x3,x4} by ENUMSET1:1;
A87: ( x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 )
proof
assume ( x1 = x3 or x1 = x4 or x2 = x3 or x2 = x4 ) ; ::_thesis: contradiction
then ( ( {A1,A5} on L1 & {A1,A5} on L2 ) or ( {A1,A5} on L1 & {A1,A5} on L2 ) or ( {A2,A5} on L1 & {A2,A5} on L2 ) or ( {A2,A5} on L1 & {A2,A5} on L2 ) ) by A5, A6, A7, A8, A9, A10, A73, A76, A84, A81, A69, A75, A83, A80, INCSP_1:1;
hence contradiction by A11, A13, A15, A16, A17, INCSP_1:def_10; ::_thesis: verum
end;
x2 in {x2} by TARSKI:def_1;
then A88: not x2 in ((A1 /\ A2) /\ A3) /\ A4 by A76, XBOOLE_0:def_5;
( k + 1 c= card (A1 \/ A2) & card (A1 \/ A2) c= k + 1 ) by A29, A23, A41, A61, A53, Th1, CARD_1:11;
then card (A1 \/ A2) = k + 1 by XBOOLE_0:def_10;
then A1 \/ A2 = L1 by A41, A19, A42, CARD_FIN:1, XBOOLE_1:8;
then A89: L1 = (((A1 /\ A2) /\ A3) /\ A4) \/ ({x1} \/ {x2}) by A73, A76, A69, A75, XBOOLE_1:5;
then A90: L1 = (((A1 /\ A2) /\ A3) /\ A4) \/ {x1,x2} by ENUMSET1:1;
A91: L1 /\ L2 c= ((A1 /\ A2) /\ A3) /\ A4
proof
assume not L1 /\ L2 c= ((A1 /\ A2) /\ A3) /\ A4 ; ::_thesis: contradiction
then consider x being set such that
A92: x in L1 /\ L2 and
A93: not x in ((A1 /\ A2) /\ A3) /\ A4 by TARSKI:def_3;
x in L1 by A92, XBOOLE_0:def_4;
then A94: x in {x1,x2} by A90, A93, XBOOLE_0:def_3;
x in L2 by A92, XBOOLE_0:def_4;
then ( x1 in L2 or x2 in L2 ) by A94, TARSKI:def_2;
then ( x1 in {x3,x4} or x2 in {x3,x4} ) by A86, A77, A88, XBOOLE_0:def_3;
hence contradiction by A87, TARSKI:def_2; ::_thesis: verum
end;
A95: ((A1 /\ A2) /\ A3) /\ A4 c= L2 by A85, XBOOLE_1:10;
((A1 /\ A2) /\ A3) /\ A4 c= L1 by A89, XBOOLE_1:10;
then ((A1 /\ A2) /\ A3) /\ A4 c= L1 /\ L2 by A95, XBOOLE_1:19;
then L1 /\ L2 = ((A1 /\ A2) /\ A3) /\ A4 by A91, XBOOLE_0:def_10;
then card k c= card (k - 1) by A22, A71, A67, A65, CARD_1:11;
then A96: k <= k - 1 by A25, NAT_1:40;
k - 1 <= (k - 1) + 1 by A25, NAT_1:11;
then k = k - 1 by A96, XXREAL_0:1;
hence contradiction ; ::_thesis: verum
end;
k - 1 c= card ((A1 /\ A2) \/ (A3 /\ A4)) by A62, CARD_1:11, XBOOLE_1:7;
then card ((A1 /\ A2) \/ (A3 /\ A4)) = k by A58, A64, XBOOLE_0:def_10;
hence A6 = (A1 /\ A2) \/ (A3 /\ A4) by A33, A34, A48, A46, CARD_FIN:1, XBOOLE_1:8; ::_thesis: verum
end;
( A6 on L2 implies A6 = (A1 /\ A2) \/ (A3 /\ A4) )
proof
assume A97: A6 on L2 ; ::_thesis: A6 = (A1 /\ A2) \/ (A3 /\ A4)
A98: A4 = A6
proof
assume A99: A4 <> A6 ; ::_thesis: contradiction
( {A4,A6} on L2 & {A4,A6} on L4 ) by A8, A14, A32, A97, INCSP_1:1;
hence contradiction by A10, A16, A99, INCSP_1:def_10; ::_thesis: verum
end;
A3 = A6
proof
assume A100: A3 <> A6 ; ::_thesis: contradiction
( {A3,A6} on L2 & {A3,A6} on L3 ) by A7, A12, A31, A97, INCSP_1:1;
hence contradiction by A10, A15, A100, INCSP_1:def_10; ::_thesis: verum
end;
hence A6 = (A1 /\ A2) \/ (A3 /\ A4) by A47, A36, A98, XBOOLE_1:12, XBOOLE_1:27; ::_thesis: verum
end;
hence A6 = (A1 /\ A2) \/ (A3 /\ A4) by A52; ::_thesis: verum
end;
( A6 on L1 implies A6 = (A1 /\ A2) \/ (A3 /\ A4) )
proof
assume A101: A6 on L1 ; ::_thesis: A6 = (A1 /\ A2) \/ (A3 /\ A4)
A102: A1 = A6
proof
assume A103: A1 <> A6 ; ::_thesis: contradiction
( {A1,A6} on L1 & {A1,A6} on L3 ) by A5, A11, A31, A101, INCSP_1:1;
hence contradiction by A9, A15, A103, INCSP_1:def_10; ::_thesis: verum
end;
A2 = A6
proof
assume A104: A2 <> A6 ; ::_thesis: contradiction
( {A2,A6} on L1 & {A2,A6} on L4 ) by A6, A13, A32, A101, INCSP_1:1;
hence contradiction by A9, A16, A104, INCSP_1:def_10; ::_thesis: verum
end;
hence A6 = (A1 /\ A2) \/ (A3 /\ A4) by A45, A36, A102, XBOOLE_1:12, XBOOLE_1:27; ::_thesis: verum
end;
hence ex A6 being POINT of (G_ (k,X)) st
( A6 on L3 & A6 on L4 & A6 = (A1 /\ A2) \/ (A3 /\ A4) ) by A31, A32, A50; ::_thesis: verum
end;
theorem :: COMBGRAS:13
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
G_ (k,X) is Desarguesian
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
G_ (k,X) is Desarguesian
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies G_ (k,X) is Desarguesian )
assume that
A1: 0 < k and
A2: k + 1 c= card X ; ::_thesis: G_ (k,X) is Desarguesian
let o, b1, a1, b2, a2, b3, a3, r, s, t be POINT of (G_ (k,X)); :: according to INCPROJ:def_13 ::_thesis: for b1, b2, b3, b4, b5, b6, b7, b8, b9 being Element of the Lines of (G_ (k,X)) holds
( not {o,b1,a1} on b1 or not {o,a2,b2} on b2 or not {o,a3,b3} on b3 or not {a3,a2,t} on b4 or not {a3,r,a1} on b5 or not {a2,s,a1} on b6 or not {t,b2,b3} on b7 or not {b1,r,b3} on b8 or not {b1,s,b2} on b9 or not b1,b2,b3 are_mutually_different or o = a1 or o = a2 or o = a3 or o = b1 or o = b2 or o = b3 or a1 = b1 or a2 = b2 or a3 = b3 or ex b10 being Element of the Lines of (G_ (k,X)) st {r,s,t} on b10 )
let C1, C2, C3, A1, A2, A3, B1, B2, B3 be LINE of (G_ (k,X)); ::_thesis: ( not {o,b1,a1} on C1 or not {o,a2,b2} on C2 or not {o,a3,b3} on C3 or not {a3,a2,t} on A1 or not {a3,r,a1} on A2 or not {a2,s,a1} on A3 or not {t,b2,b3} on B1 or not {b1,r,b3} on B2 or not {b1,s,b2} on B3 or not C1,C2,C3 are_mutually_different or o = a1 or o = a2 or o = a3 or o = b1 or o = b2 or o = b3 or a1 = b1 or a2 = b2 or a3 = b3 or ex b1 being Element of the Lines of (G_ (k,X)) st {r,s,t} on b1 )
assume that
A3: {o,b1,a1} on C1 and
A4: {o,a2,b2} on C2 and
A5: {o,a3,b3} on C3 and
A6: {a3,a2,t} on A1 and
A7: {a3,r,a1} on A2 and
A8: {a2,s,a1} on A3 and
A9: {t,b2,b3} on B1 and
A10: {b1,r,b3} on B2 and
A11: {b1,s,b2} on B3 and
A12: C1,C2,C3 are_mutually_different and
A13: o <> a1 and
A14: ( o <> a2 & o <> a3 ) and
A15: o <> b1 and
A16: ( o <> b2 & o <> b3 ) and
A17: a1 <> b1 and
A18: a2 <> b2 and
A19: a3 <> b3 ; ::_thesis: ex b1 being Element of the Lines of (G_ (k,X)) st {r,s,t} on b1
A20: o on C1 by A3, INCSP_1:2;
A21: b2 on C2 by A4, INCSP_1:2;
then A22: b2 c= C2 by A1, A2, Th10;
A23: a2 on C2 by A4, INCSP_1:2;
then A24: {a2,b2} on C2 by A21, INCSP_1:1;
A25: o on C2 by A4, INCSP_1:2;
then A26: {o,b2} on C2 by A21, INCSP_1:1;
A27: {o,a2} on C2 by A25, A23, INCSP_1:1;
A28: ( a3 on A1 & a2 on A1 ) by A6, INCSP_1:2;
A29: b3 on B2 by A10, INCSP_1:2;
A30: b3 on C3 by A5, INCSP_1:2;
then A31: b3 c= C3 by A1, A2, Th10;
A32: a3 on C3 by A5, INCSP_1:2;
then A33: {a3,b3} on C3 by A30, INCSP_1:1;
A34: o on C3 by A5, INCSP_1:2;
then A35: {o,b3} on C3 by A30, INCSP_1:1;
A36: {o,a3} on C3 by A34, A32, INCSP_1:1;
A37: ( a3 on A2 & a1 on A2 ) by A7, INCSP_1:2;
A38: b1 on B3 by A11, INCSP_1:2;
A39: C1 <> C3 by A12, ZFMISC_1:def_5;
A40: b1 on B2 by A10, INCSP_1:2;
A41: C2 <> C3 by A12, ZFMISC_1:def_5;
A42: b3 on B1 by A9, INCSP_1:2;
A43: C1 <> C2 by A12, ZFMISC_1:def_5;
A44: b2 on B1 by A9, INCSP_1:2;
A45: a1 on C1 by A3, INCSP_1:2;
then A46: a1 c= C1 by A1, A2, Th10;
A47: b1 on C1 by A3, INCSP_1:2;
then A48: {o,b1} on C1 by A20, INCSP_1:1;
A49: b2 on B3 by A11, INCSP_1:2;
A50: {a1,b1} on C1 by A47, A45, INCSP_1:1;
A51: ( not a1 on B2 & not a2 on B3 & not a3 on B1 )
proof
assume ( a1 on B2 or a2 on B3 or a3 on B1 ) ; ::_thesis: contradiction
then ( {a1,b1} on B2 or {a2,b2} on B3 or {a3,b3} on B1 ) by A42, A40, A49, INCSP_1:1;
then ( b3 on C1 or b1 on C2 or b2 on C3 ) by A17, A18, A19, A44, A29, A38, A50, A24, A33, INCSP_1:def_10;
then ( {o,b3} on C1 or {o,b1} on C2 or {o,b2} on C3 ) by A20, A25, A34, INCSP_1:1;
hence contradiction by A15, A16, A48, A26, A35, A43, A41, A39, INCSP_1:def_10; ::_thesis: verum
end;
A52: ( s on A3 & s on B3 ) by A8, A11, INCSP_1:2;
A53: t on B1 by A9, INCSP_1:2;
A54: ( r on A2 & r on B2 ) by A7, A10, INCSP_1:2;
A55: t on A1 by A6, INCSP_1:2;
A56: a1 on A3 by A8, INCSP_1:2;
A57: a2 on A3 by A8, INCSP_1:2;
A58: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A2, Def1;
A59: {o,a1} on C1 by A20, A45, INCSP_1:1;
A60: ( not o on A1 & not o on B1 & not o on A2 & not o on B2 & not o on A3 & not o on B3 )
proof
assume ( o on A1 or o on B1 or o on A2 or o on B2 or o on A3 or o on B3 ) ; ::_thesis: contradiction
then ( ( {o,a2} on A1 & {o,a3} on A1 ) or ( {o,b2} on B1 & {o,b3} on B1 ) or ( {o,a1} on A2 & {o,a3} on A2 ) or ( {o,b1} on B2 & {o,b3} on B2 ) or ( {o,a2} on A3 & {o,a1} on A3 ) or ( {o,b2} on B3 & {o,b1} on B3 ) ) by A28, A37, A57, A56, A44, A42, A40, A29, A38, A49, INCSP_1:1;
then ( ( A1 = C2 & A1 = C3 ) or ( B1 = C2 & B1 = C3 ) or ( A2 = C1 & A2 = C3 ) or ( B2 = C1 & B2 = C3 ) or ( A3 = C2 & A3 = C1 ) or ( B3 = C2 & B3 = C1 ) ) by A13, A14, A15, A16, A59, A27, A36, A48, A26, A35, INCSP_1:def_10;
hence contradiction by A12, ZFMISC_1:def_5; ::_thesis: verum
end;
then consider salpha being POINT of (G_ (k,X)) such that
A61: ( salpha on A3 & salpha on B3 ) and
A62: salpha = (a1 /\ b1) \/ (a2 /\ b2) by A1, A2, A20, A47, A45, A25, A23, A21, A57, A56, A38, A49, A43, A51, Th12;
consider ralpha being POINT of (G_ (k,X)) such that
A63: ( ralpha on B2 & ralpha on A2 ) and
A64: ralpha = (a1 /\ b1) \/ (a3 /\ b3) by A1, A2, A20, A47, A45, A34, A32, A30, A37, A40, A29, A39, A51, A60, Th12;
A65: ((a1 /\ b1) \/ (a3 /\ b3)) \/ (a2 /\ b2) = (a1 /\ b1) \/ ((a3 /\ b3) \/ (a2 /\ b2)) by XBOOLE_1:4;
consider talpha being POINT of (G_ (k,X)) such that
A66: ( talpha on A1 & talpha on B1 ) and
A67: talpha = (a2 /\ b2) \/ (a3 /\ b3) by A1, A2, A25, A23, A21, A34, A32, A30, A28, A44, A42, A41, A51, A60, Th12;
A68: ( A1 <> B1 & A2 <> B2 ) by A6, A7, A51, INCSP_1:2;
A69: ( r = ralpha & s = salpha & t = talpha )
proof
A70: ( {s,salpha} on A3 & {s,salpha} on B3 ) by A52, A61, INCSP_1:1;
A71: ( {r,ralpha} on A2 & {r,ralpha} on B2 ) by A54, A63, INCSP_1:1;
assume A72: ( r <> ralpha or s <> salpha or t <> talpha ) ; ::_thesis: contradiction
( {t,talpha} on A1 & {t,talpha} on B1 ) by A55, A53, A66, INCSP_1:1;
hence contradiction by A57, A51, A68, A72, A71, A70, INCSP_1:def_10; ::_thesis: verum
end;
then r \/ s = (((a3 /\ b3) \/ (a1 /\ b1)) \/ (a1 /\ b1)) \/ (a2 /\ b2) by A62, A64, XBOOLE_1:4;
then r \/ s = ((a3 /\ b3) \/ ((a1 /\ b1) \/ (a1 /\ b1))) \/ (a2 /\ b2) by XBOOLE_1:4;
then A73: (r \/ s) \/ t = ((a1 /\ b1) \/ (a3 /\ b3)) \/ (a2 /\ b2) by A67, A69, A65, XBOOLE_1:7, XBOOLE_1:12;
a2 c= C2 by A1, A2, A23, Th10;
then A74: a2 \/ b2 c= C2 by A22, XBOOLE_1:8;
r c= r \/ (s \/ t) by XBOOLE_1:7;
then A75: r c= (r \/ s) \/ t by XBOOLE_1:4;
C1 in the Lines of (G_ (k,X)) ;
then A76: ex C11 being Subset of X st
( C11 = C1 & card C11 = k + 1 ) by A58;
A77: b1 c= C1 by A1, A2, A47, Th10;
then a1 \/ b1 c= C1 by A46, XBOOLE_1:8;
then A78: card (a1 \/ b1) c= k + 1 by A76, CARD_1:11;
A79: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, A2, Def1;
o in the Points of (G_ (k,X)) ;
then A80: ex o1 being Subset of X st
( o1 = o & card o1 = k ) by A79;
b1 in the Points of (G_ (k,X)) ;
then A81: ex b11 being Subset of X st
( b11 = b1 & card b11 = k ) by A79;
a3 in the Points of (G_ (k,X)) ;
then A82: ex a13 being Subset of X st
( a13 = a3 & card a13 = k ) by A79;
then A83: card a3 = (k - 1) + 1 ;
t in the Points of (G_ (k,X)) ;
then A84: ex t1 being Subset of X st
( t1 = t & card t1 = k ) by A79;
then A85: t is finite ;
a2 in the Points of (G_ (k,X)) ;
then A86: ex a12 being Subset of X st
( a12 = a2 & card a12 = k ) by A79;
then A87: card a2 = (k - 1) + 1 ;
s in the Points of (G_ (k,X)) ;
then A88: ex s1 being Subset of X st
( s1 = s & card s1 = k ) by A79;
then A89: s is finite ;
a1 in the Points of (G_ (k,X)) ;
then A90: ex a11 being Subset of X st
( a11 = a1 & card a11 = k ) by A79;
then k + 1 c= card (a1 \/ b1) by A81, A17, Th1;
then A91: card (a1 \/ b1) = (k - 1) + (2 * 1) by A78, XBOOLE_0:def_10;
A92: k - 1 is Element of NAT by A1, NAT_1:20;
C2 in the Lines of (G_ (k,X)) ;
then ex C12 being Subset of X st
( C12 = C2 & card C12 = k + 1 ) by A58;
then A93: card (a2 \/ b2) c= k + 1 by A74, CARD_1:11;
b2 in the Points of (G_ (k,X)) ;
then A94: ex b12 being Subset of X st
( b12 = b2 & card b12 = k ) by A79;
then k + 1 c= card (a2 \/ b2) by A86, A18, Th1;
then A95: card (a2 \/ b2) = (k - 1) + (2 * 1) by A93, XBOOLE_0:def_10;
then A96: card (a2 /\ b2) = k - 1 by A92, A94, A87, Th2;
A97: card (a2 /\ b2) = (k - 2) + 1 by A92, A94, A95, A87, Th2;
A98: card a1 = (k - 1) + 1 by A90;
then A99: card (a1 /\ b1) = k - 1 by A92, A81, A91, Th2;
a3 c= C3 by A1, A2, A32, Th10;
then A100: a3 \/ b3 c= C3 by A31, XBOOLE_1:8;
s c= s \/ (r \/ t) by XBOOLE_1:7;
then A101: s c= (r \/ s) \/ t by XBOOLE_1:4;
0 + 1 < k + 1 by A1, XREAL_1:8;
then 1 <= (k - 1) + 1 by NAT_1:13;
then ( 1 <= k - 1 or k = 1 ) by A92, NAT_1:8;
then A102: ( 1 + 1 <= (k - 1) + 1 or k = 1 ) by XREAL_1:6;
A103: o c= C1 by A1, A2, A20, Th10;
A104: not k = 1
proof
assume A105: k = 1 ; ::_thesis: contradiction
then consider z1 being set such that
A106: o = {z1} by A80, CARD_2:42;
consider z3 being set such that
A107: b1 = {z3} by A81, A105, CARD_2:42;
consider z2 being set such that
A108: a1 = {z2} by A90, A105, CARD_2:42;
o \/ a1 c= C1 by A103, A46, XBOOLE_1:8;
then (o \/ a1) \/ b1 c= C1 by A77, XBOOLE_1:8;
then {z1,z2} \/ b1 c= C1 by A106, A108, ENUMSET1:1;
then A109: {z1,z2,z3} c= C1 by A107, ENUMSET1:3;
card {z1,z2,z3} = 3 by A13, A15, A17, A106, A108, A107, CARD_2:58;
then 3 c= card C1 by A109, CARD_1:11;
hence contradiction by A76, A105, NAT_1:39; ::_thesis: verum
end;
then A110: k - 2 is Element of NAT by A102, NAT_1:21;
C3 in the Lines of (G_ (k,X)) ;
then ex C13 being Subset of X st
( C13 = C3 & card C13 = k + 1 ) by A58;
then A111: card (a3 \/ b3) c= k + 1 by A100, CARD_1:11;
b3 in the Points of (G_ (k,X)) ;
then A112: ex b13 being Subset of X st
( b13 = b3 & card b13 = k ) by A79;
then k + 1 c= card (a3 \/ b3) by A82, A19, Th1;
then A113: card (a3 \/ b3) = (k - 1) + (2 * 1) by A111, XBOOLE_0:def_10;
then A114: card (a3 /\ b3) = k - 1 by A92, A112, A83, Th2;
r in the Points of (G_ (k,X)) ;
then A115: ex r1 being Subset of X st
( r1 = r & card r1 = k ) by A79;
then r \/ s c= X by A88, XBOOLE_1:8;
then A116: (r \/ s) \/ t c= X by A84, XBOOLE_1:8;
A117: card (a3 /\ b3) = (k - 2) + 1 by A92, A112, A113, A83, Th2;
A118: card (a1 /\ b1) = (k - 2) + 1 by A92, A81, A91, A98, Th2;
card ((a1 /\ b1) \/ (a2 /\ b2)) = (k - 2) + (2 * 1) by A62, A69, A88;
then A119: card ((a1 /\ b1) /\ (a2 /\ b2)) = k - 2 by A110, A118, A97, Th2;
card ((a2 /\ b2) \/ (a3 /\ b3)) = (k - 2) + (2 * 1) by A67, A69, A84;
then A120: card ((a2 /\ b2) /\ (a3 /\ b3)) = k - 2 by A110, A97, A117, Th2;
card ((a1 /\ b1) \/ (a3 /\ b3)) = (k - 2) + (2 * 1) by A64, A69, A115;
then A121: card ((a1 /\ b1) /\ (a3 /\ b3)) = k - 2 by A110, A118, A117, Th2;
A122: t c= (r \/ s) \/ t by XBOOLE_1:7;
A123: ( k = 2 implies ex O being LINE of (G_ (k,X)) st {r,s,t} on O )
proof
assume k = 2 ; ::_thesis: ex O being LINE of (G_ (k,X)) st {r,s,t} on O
then card ((r \/ s) \/ t) = k + 1 by A99, A96, A114, A73, A121, A120, A119, Th7;
then (r \/ s) \/ t in the Lines of (G_ (k,X)) by A58, A116;
then consider O being LINE of (G_ (k,X)) such that
A124: O = (r \/ s) \/ t ;
A125: t on O by A1, A2, A122, A124, Th10;
( r on O & s on O ) by A1, A2, A75, A101, A124, Th10;
then {r,s,t} on O by A125, INCSP_1:2;
hence ex O being LINE of (G_ (k,X)) st {r,s,t} on O ; ::_thesis: verum
end;
A126: r is finite by A115;
A127: ( 3 <= k implies ex O being LINE of (G_ (k,X)) st {r,s,t} on O )
proof
A128: ( card ((r \/ s) \/ t) = k + 1 implies ex O being LINE of (G_ (k,X)) st {r,s,t} on O )
proof
assume card ((r \/ s) \/ t) = k + 1 ; ::_thesis: ex O being LINE of (G_ (k,X)) st {r,s,t} on O
then (r \/ s) \/ t in the Lines of (G_ (k,X)) by A58, A116;
then consider O being LINE of (G_ (k,X)) such that
A129: O = (r \/ s) \/ t ;
A130: t on O by A1, A2, A122, A129, Th10;
( r on O & s on O ) by A1, A2, A75, A101, A129, Th10;
then {r,s,t} on O by A130, INCSP_1:2;
hence ex O being LINE of (G_ (k,X)) st {r,s,t} on O ; ::_thesis: verum
end;
A131: ( card ((r \/ s) \/ t) = k implies ex O being LINE of (G_ (k,X)) st {r,s,t} on O )
proof
assume A132: card ((r \/ s) \/ t) = k ; ::_thesis: ex O being LINE of (G_ (k,X)) st {r,s,t} on O
then A133: t = (r \/ s) \/ t by A84, A126, A89, A85, CARD_FIN:1, XBOOLE_1:7;
( r = (r \/ s) \/ t & s = (r \/ s) \/ t ) by A115, A88, A126, A89, A85, A75, A101, A132, CARD_FIN:1;
then {r,s,t} on A1 by A55, A133, INCSP_1:2;
hence ex O being LINE of (G_ (k,X)) st {r,s,t} on O ; ::_thesis: verum
end;
assume 3 <= k ; ::_thesis: ex O being LINE of (G_ (k,X)) st {r,s,t} on O
hence ex O being LINE of (G_ (k,X)) st {r,s,t} on O by A99, A96, A114, A73, A102, A121, A120, A119, A128, A131, Th7; ::_thesis: verum
end;
( k = 2 or 2 <= k - 1 ) by A92, A104, A102, NAT_1:8;
then ( k = 2 or 2 + 1 <= (k - 1) + 1 ) by XREAL_1:6;
hence ex b1 being Element of the Lines of (G_ (k,X)) st {r,s,t} on b1 by A123, A127; ::_thesis: verum
end;
definition
let S be IncProjStr ;
let K be Subset of the Points of S;
attrK is clique means :Def2: :: COMBGRAS:def 2
for A, B being POINT of S st A in K & B in K holds
ex L being LINE of S st {A,B} on L;
end;
:: deftheorem Def2 defines clique COMBGRAS:def_2_:_
for S being IncProjStr
for K being Subset of the Points of S holds
( K is clique iff for A, B being POINT of S st A in K & B in K holds
ex L being LINE of S st {A,B} on L );
definition
let S be IncProjStr ;
let K be Subset of the Points of S;
attrK is maximal_clique means :Def3: :: COMBGRAS:def 3
( K is clique & ( for U being Subset of the Points of S st U is clique & K c= U holds
U = K ) );
end;
:: deftheorem Def3 defines maximal_clique COMBGRAS:def_3_:_
for S being IncProjStr
for K being Subset of the Points of S holds
( K is maximal_clique iff ( K is clique & ( for U being Subset of the Points of S st U is clique & K c= U holds
U = K ) ) );
definition
let k be Element of NAT ;
let X be non empty set ;
let T be Subset of the Points of (G_ (k,X));
attrT is STAR means :Def4: :: COMBGRAS:def 4
ex S being Subset of X st
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } );
attrT is TOP means :Def5: :: COMBGRAS:def 5
ex S being Subset of X st
( card S = k + 1 & T = { A where A is Subset of X : ( card A = k & A c= S ) } );
end;
:: deftheorem Def4 defines STAR COMBGRAS:def_4_:_
for k being Element of NAT
for X being non empty set
for T being Subset of the Points of (G_ (k,X)) holds
( T is STAR iff ex S being Subset of X st
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } ) );
:: deftheorem Def5 defines TOP COMBGRAS:def_5_:_
for k being Element of NAT
for X being non empty set
for T being Subset of the Points of (G_ (k,X)) holds
( T is TOP iff ex S being Subset of X st
( card S = k + 1 & T = { A where A is Subset of X : ( card A = k & A c= S ) } ) );
theorem Th14: :: COMBGRAS:14
for k being Element of NAT
for X being non empty set st 2 <= k & k + 2 c= card X holds
for K being Subset of the Points of (G_ (k,X)) st ( K is STAR or K is TOP ) holds
K is maximal_clique
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 2 <= k & k + 2 c= card X holds
for K being Subset of the Points of (G_ (k,X)) st ( K is STAR or K is TOP ) holds
K is maximal_clique
let X be non empty set ; ::_thesis: ( 2 <= k & k + 2 c= card X implies for K being Subset of the Points of (G_ (k,X)) st ( K is STAR or K is TOP ) holds
K is maximal_clique )
assume that
A1: 2 <= k and
A2: k + 2 c= card X ; ::_thesis: for K being Subset of the Points of (G_ (k,X)) st ( K is STAR or K is TOP ) holds
K is maximal_clique
A3: k - 2 is Element of NAT by A1, NAT_1:21;
let K be Subset of the Points of (G_ (k,X)); ::_thesis: ( ( K is STAR or K is TOP ) implies K is maximal_clique )
A4: succ k = k + 1 by NAT_1:38;
A5: succ (k + 1) = (k + 1) + 1 by NAT_1:38;
k + 1 <= k + 2 by XREAL_1:7;
then k + 1 c= k + 2 by NAT_1:39;
then A6: k + 1 c= card X by A2, XBOOLE_1:1;
then A7: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, Def1;
A8: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A6, Def1;
A9: k - 1 is Element of NAT by A1, NAT_1:21, XXREAL_0:2;
then A10: succ (k - 1) = (k - 1) + 1 by NAT_1:38;
A11: ( K is STAR implies K is maximal_clique )
proof
assume K is STAR ; ::_thesis: K is maximal_clique
then consider S being Subset of X such that
A12: card S = k - 1 and
A13: K = { A where A is Subset of X : ( card A = k & S c= A ) } by Def4;
A14: S is finite by A1, A12, NAT_1:21, XXREAL_0:2;
A15: for U being Subset of the Points of (G_ (k,X)) st U is clique & K c= U holds
U = K
proof
A16: succ (k - 2) = (k - 2) + 1 by A3, NAT_1:38;
let U be Subset of the Points of (G_ (k,X)); ::_thesis: ( U is clique & K c= U implies U = K )
assume that
A17: U is clique and
A18: K c= U and
A19: U <> K ; ::_thesis: contradiction
not U c= K by A18, A19, XBOOLE_0:def_10;
then consider A being set such that
A20: A in U and
A21: not A in K by TARSKI:def_3;
consider A2 being POINT of (G_ (k,X)) such that
A22: A2 = A by A20;
card (S /\ A) c= k - 1 by A12, CARD_1:11, XBOOLE_1:17;
then card (S /\ A) in succ (k - 1) by A9, ORDINAL1:22;
then A23: ( card (S /\ A) in succ (k - 2) or card (S /\ A) = k - 1 ) by A16, ORDINAL1:8;
A24: ( S /\ A c= S & S /\ A c= A ) by XBOOLE_1:17;
A in the Points of (G_ (k,X)) by A20;
then A25: ex A1 being Subset of X st
( A1 = A & card A1 = k ) by A7;
then not S c= A by A13, A21;
then A26: card (S /\ A) c= k - 2 by A3, A12, A14, A24, A23, CARD_FIN:1, ORDINAL1:22;
A27: not X \ (A \/ S) <> {}
proof
A28: succ (k - 2) = (k - 2) + 1 by A3, NAT_1:38;
assume X \ (A \/ S) <> {} ; ::_thesis: contradiction
then consider y being set such that
A29: y in X \ (A \/ S) by XBOOLE_0:def_1;
A30: not y in A \/ S by A29, XBOOLE_0:def_5;
then A31: not y in S by XBOOLE_0:def_3;
then A32: card (S \/ {y}) = (k - 1) + 1 by A12, A14, CARD_2:41;
A33: {y} c= X by A29, ZFMISC_1:31;
then S \/ {y} c= X by XBOOLE_1:8;
then S \/ {y} in the Points of (G_ (k,X)) by A7, A32;
then consider B being POINT of (G_ (k,X)) such that
A34: B = S \/ {y} ;
A35: not y in A by A30, XBOOLE_0:def_3;
A /\ B c= A /\ S
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A /\ B or a in A /\ S )
assume A36: a in A /\ B ; ::_thesis: a in A /\ S
then a in S \/ {y} by A34, XBOOLE_0:def_4;
then A37: ( a in S or a in {y} ) by XBOOLE_0:def_3;
a in A by A36, XBOOLE_0:def_4;
hence a in A /\ S by A35, A37, TARSKI:def_1, XBOOLE_0:def_4; ::_thesis: verum
end;
then card (A /\ B) c= card (A /\ S) by CARD_1:11;
then card (A /\ B) c= k - 2 by A26, XBOOLE_1:1;
then A38: card (A /\ B) in k - 1 by A3, A28, ORDINAL1:22;
A39: for L being LINE of (G_ (k,X)) holds not {A2,B} on L
proof
A <> B
proof
assume A = B ; ::_thesis: contradiction
then {y} c= A by A34, XBOOLE_1:7;
hence contradiction by A35, ZFMISC_1:31; ::_thesis: verum
end;
then A40: k + 1 c= card (A \/ B) by A25, A32, A34, Th1;
assume ex L being LINE of (G_ (k,X)) st {A2,B} on L ; ::_thesis: contradiction
then consider L being LINE of (G_ (k,X)) such that
A41: {A2,B} on L ;
B on L by A41, INCSP_1:1;
then A42: B c= L by A1, A6, Th10;
L in the Lines of (G_ (k,X)) ;
then A43: ex L1 being Subset of X st
( L = L1 & card L1 = k + 1 ) by A8;
A2 on L by A41, INCSP_1:1;
then A c= L by A1, A6, A22, Th10;
then A \/ B c= L by A42, XBOOLE_1:8;
then card (A \/ B) c= k + 1 by A43, CARD_1:11;
then A44: card (A \/ B) = (k - 1) + (2 * 1) by A40, XBOOLE_0:def_10;
card B = (k - 1) + 1 by A12, A14, A31, A34, CARD_2:41;
then card (A /\ B) = k - 1 by A9, A25, A44, Th2;
hence contradiction by A38; ::_thesis: verum
end;
A45: S c= B by A34, XBOOLE_1:7;
B c= X by A33, A34, XBOOLE_1:8;
then B in K by A13, A32, A34, A45;
hence contradiction by A17, A18, A20, A22, A39, Def2; ::_thesis: verum
end;
k - 1 < (k - 1) + 1 by A9, NAT_1:13;
then card S in card A by A9, A12, A25, NAT_1:44;
then A \ S <> {} by CARD_1:68;
then consider x being set such that
A46: x in A \ S by XBOOLE_0:def_1;
not x in S by A46, XBOOLE_0:def_5;
then A47: card (S \/ {x}) = (k - 1) + 1 by A12, A14, CARD_2:41;
A48: {x} c= A by A46, ZFMISC_1:31;
x in A by A46;
then A49: {x} c= X by A25, ZFMISC_1:31;
then A50: S \/ {x} c= X by XBOOLE_1:8;
not X \ (A \/ S) = {}
proof
assume X \ (A \/ S) = {} ; ::_thesis: contradiction
then A51: X c= A \/ S by XBOOLE_1:37;
S \/ {x} in the Points of (G_ (k,X)) by A7, A47, A50;
then consider B being POINT of (G_ (k,X)) such that
A52: B = S \/ {x} ;
A \/ B = (A \/ S) \/ {x} by A52, XBOOLE_1:4;
then A53: A \/ B = A \/ S by A48, XBOOLE_1:10, XBOOLE_1:12;
A \/ S c= X by A25, XBOOLE_1:8;
then A54: A \/ S = X by A51, XBOOLE_0:def_10;
A55: for L being LINE of (G_ (k,X)) holds not {A2,B} on L
proof
assume ex L being LINE of (G_ (k,X)) st {A2,B} on L ; ::_thesis: contradiction
then consider L being LINE of (G_ (k,X)) such that
A56: {A2,B} on L ;
B on L by A56, INCSP_1:1;
then A57: B c= L by A1, A6, Th10;
A2 on L by A56, INCSP_1:1;
then A c= L by A1, A6, A22, Th10;
then A \/ B c= L by A57, XBOOLE_1:8;
then card (A \/ B) c= card L by CARD_1:11;
then A58: k + 2 c= card L by A2, A54, A53, XBOOLE_1:1;
L in the Lines of (G_ (k,X)) ;
then ex L1 being Subset of X st
( L = L1 & card L1 = k + 1 ) by A8;
then k + 1 in k + 1 by A5, A58, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
( S c= B & B c= X ) by A49, A52, XBOOLE_1:8, XBOOLE_1:10;
then B in K by A13, A47, A52;
hence contradiction by A17, A18, A20, A22, A55, Def2; ::_thesis: verum
end;
hence contradiction by A27; ::_thesis: verum
end;
K is clique
proof
let A, B be POINT of (G_ (k,X)); :: according to COMBGRAS:def_2 ::_thesis: ( A in K & B in K implies ex L being LINE of (G_ (k,X)) st {A,B} on L )
assume that
A59: A in K and
A60: B in K ; ::_thesis: ex L being LINE of (G_ (k,X)) st {A,B} on L
A61: ex A1 being Subset of X st
( A1 = A & card A1 = k & S c= A1 ) by A13, A59;
then A62: A is finite ;
A63: ex B1 being Subset of X st
( B1 = B & card B1 = k & S c= B1 ) by A13, A60;
then S c= A /\ B by A61, XBOOLE_1:19;
then k - 1 c= card (A /\ B) by A12, CARD_1:11;
then k - 1 in succ (card (A /\ B)) by A9, ORDINAL1:22;
then ( card (A /\ B) = k - 1 or k - 1 in card (A /\ B) ) by ORDINAL1:8;
then A64: ( card (A /\ B) = k - 1 or k c= card (A /\ B) ) by A10, ORDINAL1:21;
A65: B is finite by A63;
A66: ( card (A /\ B) = k implies ex L being LINE of (G_ (k,X)) st {A,B} on L )
proof
A67: card A <> card X
proof
assume card A = card X ; ::_thesis: contradiction
then k in k by A6, A4, A61, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
card A c= card X by A61, CARD_1:11;
then card A in card X by A67, CARD_1:3;
then X \ A <> {} by CARD_1:68;
then consider x being set such that
A68: x in X \ A by XBOOLE_0:def_1;
{x} c= X by A68, ZFMISC_1:31;
then A69: A \/ {x} c= X by A61, XBOOLE_1:8;
not x in A by A68, XBOOLE_0:def_5;
then card (A \/ {x}) = k + 1 by A61, A62, CARD_2:41;
then A \/ {x} in the Lines of (G_ (k,X)) by A8, A69;
then consider L being LINE of (G_ (k,X)) such that
A70: L = A \/ {x} ;
assume card (A /\ B) = k ; ::_thesis: ex L being LINE of (G_ (k,X)) st {A,B} on L
then ( A /\ B = A & A /\ B = B ) by A61, A63, A62, A65, CARD_FIN:1, XBOOLE_1:17;
then B c= A \/ {x} by XBOOLE_1:7;
then A71: B on L by A1, A6, A70, Th10;
A c= A \/ {x} by XBOOLE_1:7;
then A on L by A1, A6, A70, Th10;
then {A,B} on L by A71, INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {A,B} on L ; ::_thesis: verum
end;
A72: ( card (A /\ B) = k - 1 implies ex L being LINE of (G_ (k,X)) st {A,B} on L )
proof
A73: A \/ B c= X by A61, A63, XBOOLE_1:8;
assume A74: card (A /\ B) = k - 1 ; ::_thesis: ex L being LINE of (G_ (k,X)) st {A,B} on L
card A = (k - 1) + 1 by A61;
then card (A \/ B) = (k - 1) + (2 * 1) by A9, A63, A74, Th2;
then A \/ B in the Lines of (G_ (k,X)) by A8, A73;
then consider L being LINE of (G_ (k,X)) such that
A75: L = A \/ B ;
B c= A \/ B by XBOOLE_1:7;
then A76: B on L by A1, A6, A75, Th10;
A c= A \/ B by XBOOLE_1:7;
then A on L by A1, A6, A75, Th10;
then {A,B} on L by A76, INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {A,B} on L ; ::_thesis: verum
end;
card (A /\ B) c= k by A61, CARD_1:11, XBOOLE_1:17;
hence ex L being LINE of (G_ (k,X)) st {A,B} on L by A64, A72, A66, XBOOLE_0:def_10; ::_thesis: verum
end;
hence K is maximal_clique by A15, Def3; ::_thesis: verum
end;
A77: succ 0 = 0 + 1 ;
( K is TOP implies K is maximal_clique )
proof
assume K is TOP ; ::_thesis: K is maximal_clique
then consider S being Subset of X such that
A78: card S = k + 1 and
A79: K = { A where A is Subset of X : ( card A = k & A c= S ) } by Def5;
A80: S is finite by A78;
A81: for U being Subset of the Points of (G_ (k,X)) st U is clique & K c= U holds
U = K
proof
A82: k - 2 <= (k - 2) + 1 by A3, NAT_1:11;
let U be Subset of the Points of (G_ (k,X)); ::_thesis: ( U is clique & K c= U implies U = K )
assume that
A83: U is clique and
A84: K c= U and
A85: U <> K ; ::_thesis: contradiction
not U c= K by A84, A85, XBOOLE_0:def_10;
then consider A being set such that
A86: A in U and
A87: not A in K by TARSKI:def_3;
consider A2 being POINT of (G_ (k,X)) such that
A88: A2 = A by A86;
A in the Points of (G_ (k,X)) by A86;
then A89: ex A1 being Subset of X st
( A1 = A & card A1 = k ) by A7;
then A90: A is finite ;
A91: card A <> card S by A78, A89;
A92: not A c= S by A79, A87, A89;
then consider x being set such that
A93: x in A and
A94: not x in S by TARSKI:def_3;
k <= k + 1 by NAT_1:11;
then card A c= card S by A78, A89, NAT_1:39;
then card A in card S by A91, CARD_1:3;
then A95: S \ A <> {} by CARD_1:68;
2 c= card (S \ A)
proof
A96: not card (S \ A) = 1
proof
assume card (S \ A) = 1 ; ::_thesis: contradiction
then A97: card (S \ (S \ A)) = (k + 1) - 1 by A78, A80, CARD_2:44, XBOOLE_1:36;
( S \ (S \ A) = S /\ A & S /\ A c= S ) by XBOOLE_1:17, XBOOLE_1:48;
hence contradiction by A89, A92, A90, A97, CARD_FIN:1, XBOOLE_1:17; ::_thesis: verum
end;
assume not 2 c= card (S \ A) ; ::_thesis: contradiction
then card (S \ A) in succ 1 by ORDINAL1:16;
then ( card (S \ A) in 1 or card (S \ A) = 1 ) by ORDINAL1:8;
then ( card (S \ A) c= 0 or card (S \ A) = 1 ) by A77, ORDINAL1:22;
hence contradiction by A95, A96; ::_thesis: verum
end;
then consider B1 being set such that
A98: B1 c= S \ A and
A99: card B1 = 2 by CARD_FIL:36;
A100: B1 c= S by A98, XBOOLE_1:106;
then A101: not x in B1 by A94;
card (S \ B1) = (k + 1) - 2 by A78, A80, A98, A99, CARD_2:44, XBOOLE_1:106;
then k - 2 c= card (S \ B1) by A3, A82, NAT_1:39;
then consider B2 being set such that
A102: B2 c= S \ B1 and
A103: card B2 = k - 2 by A3, CARD_FIL:36;
A104: card (B1 \/ B2) = 2 + (k - 2) by A80, A98, A99, A102, A103, CARD_2:40, XBOOLE_1:106;
A105: B2 c= X by A102, XBOOLE_1:1;
B1 c= X by A98, XBOOLE_1:1;
then A106: B1 \/ B2 c= X by A105, XBOOLE_1:8;
then B1 \/ B2 in the Points of (G_ (k,X)) by A7, A104;
then consider B being POINT of (G_ (k,X)) such that
A107: B = B1 \/ B2 ;
B1 misses A by A98, XBOOLE_1:106;
then A108: B1 /\ A = {} by XBOOLE_0:def_7;
B2 c= S by A102, XBOOLE_1:106;
then A109: B1 \/ B2 c= S by A100, XBOOLE_1:8;
then A110: not x in B1 \/ B2 by A94;
A111: A /\ B c= A \/ B by XBOOLE_1:29;
A112: k + 2 c= card (A \/ B)
proof
A113: {x} \/ B1 misses A /\ B
proof
assume not {x} \/ B1 misses A /\ B ; ::_thesis: contradiction
then ({x} \/ B1) /\ (A /\ B) <> {} by XBOOLE_0:def_7;
then consider y being set such that
A114: y in ({x} \/ B1) /\ (A /\ B) by XBOOLE_0:def_1;
y in A /\ B by A114, XBOOLE_0:def_4;
then A115: ( y in A & y in B ) by XBOOLE_0:def_4;
y in {x} \/ B1 by A114, XBOOLE_0:def_4;
then ( y in {x} or y in B1 ) by XBOOLE_0:def_3;
hence contradiction by A107, A110, A108, A115, TARSKI:def_1, XBOOLE_0:def_4; ::_thesis: verum
end;
{x} c= A by A93, ZFMISC_1:31;
then {x} c= A \/ B by XBOOLE_1:10;
then A116: (A /\ B) \/ {x} c= A \/ B by A111, XBOOLE_1:8;
B1 c= B by A107, XBOOLE_1:10;
then B1 c= A \/ B by XBOOLE_1:10;
then ((A /\ B) \/ {x}) \/ B1 c= A \/ B by A116, XBOOLE_1:8;
then (A /\ B) \/ ({x} \/ B1) c= A \/ B by XBOOLE_1:4;
then A117: card ((A /\ B) \/ ({x} \/ B1)) c= card (A \/ B) by CARD_1:11;
assume not k + 2 c= card (A \/ B) ; ::_thesis: contradiction
then A118: card (A \/ B) in succ (k + 1) by A5, ORDINAL1:16;
then A119: card (A \/ B) c= k + 1 by ORDINAL1:22;
( card (A \/ B) = k + 1 or ( card (A \/ B) in succ k & A c= A \/ B ) ) by A4, A118, ORDINAL1:8, XBOOLE_1:7;
then ( card (A \/ B) = k + 1 or ( card (A \/ B) c= k & k c= card (A \/ B) ) ) by A89, CARD_1:11, ORDINAL1:22;
then A120: ( card (A \/ B) = (k - 1) + (2 * 1) or card (A \/ B) = k + (2 * 0) ) by XBOOLE_0:def_10;
card A = (k - 1) + 1 by A89;
then A121: ( card (A /\ B) = k - 1 or card (A /\ B) = k ) by A9, A104, A107, A120, Th2;
card ({x} \/ B1) = 2 + 1 by A80, A98, A99, A101, CARD_2:41;
then ( card ((A /\ B) \/ ({x} \/ B1)) = (k - 1) + 3 or card ((A /\ B) \/ ({x} \/ B1)) = k + 3 ) by A80, A90, A98, A113, A121, CARD_2:40;
then ( k + 2 c= k + 1 or k + 3 c= k + 1 ) by A119, A117, XBOOLE_1:1;
then ( k + 1 in k + 1 or k + 3 <= k + 1 ) by A5, NAT_1:39, ORDINAL1:21;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
A122: for L being LINE of (G_ (k,X)) holds not {A2,B} on L
proof
assume ex L being LINE of (G_ (k,X)) st {A2,B} on L ; ::_thesis: contradiction
then consider L being LINE of (G_ (k,X)) such that
A123: {A2,B} on L ;
B on L by A123, INCSP_1:1;
then A124: B c= L by A1, A6, Th10;
L in the Lines of (G_ (k,X)) ;
then A125: ex L1 being Subset of X st
( L = L1 & card L1 = k + 1 ) by A8;
A2 on L by A123, INCSP_1:1;
then A c= L by A1, A6, A88, Th10;
then A \/ B c= L by A124, XBOOLE_1:8;
then A126: card (A \/ B) c= k + 1 by A125, CARD_1:11;
k + 1 c= card (A \/ B) by A89, A93, A104, A107, A110, Th1;
then k + 2 c= k + 1 by A112, A126, XBOOLE_0:def_10;
then k + 1 in k + 1 by A5, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
B in K by A79, A104, A106, A109, A107;
hence contradiction by A83, A84, A86, A88, A122, Def2; ::_thesis: verum
end;
K is clique
proof
let A, B be POINT of (G_ (k,X)); :: according to COMBGRAS:def_2 ::_thesis: ( A in K & B in K implies ex L being LINE of (G_ (k,X)) st {A,B} on L )
assume that
A127: A in K and
A128: B in K ; ::_thesis: ex L being LINE of (G_ (k,X)) st {A,B} on L
S in the Lines of (G_ (k,X)) by A8, A78;
then consider L being LINE of (G_ (k,X)) such that
A129: L = S ;
ex B1 being Subset of X st
( B1 = B & card B1 = k & B1 c= S ) by A79, A128;
then A130: B on L by A1, A6, A129, Th10;
ex A1 being Subset of X st
( A1 = A & card A1 = k & A1 c= S ) by A79, A127;
then A on L by A1, A6, A129, Th10;
then {A,B} on L by A130, INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {A,B} on L ; ::_thesis: verum
end;
hence K is maximal_clique by A81, Def3; ::_thesis: verum
end;
hence ( ( K is STAR or K is TOP ) implies K is maximal_clique ) by A11; ::_thesis: verum
end;
theorem Th15: :: COMBGRAS:15
for k being Element of NAT
for X being non empty set st 2 <= k & k + 2 c= card X holds
for K being Subset of the Points of (G_ (k,X)) holds
( not K is maximal_clique or K is STAR or K is TOP )
proof
A1: succ 0 = 0 + 1 ;
A2: succ 2 = 2 + 1 ;
let k be Element of NAT ; ::_thesis: for X being non empty set st 2 <= k & k + 2 c= card X holds
for K being Subset of the Points of (G_ (k,X)) holds
( not K is maximal_clique or K is STAR or K is TOP )
let X be non empty set ; ::_thesis: ( 2 <= k & k + 2 c= card X implies for K being Subset of the Points of (G_ (k,X)) holds
( not K is maximal_clique or K is STAR or K is TOP ) )
assume that
A3: 2 <= k and
A4: k + 2 c= card X ; ::_thesis: for K being Subset of the Points of (G_ (k,X)) holds
( not K is maximal_clique or K is STAR or K is TOP )
k + 1 <= k + 2 by XREAL_1:7;
then A5: k + 1 c= k + 2 by NAT_1:39;
then A6: k + 1 c= card X by A4, XBOOLE_1:1;
then A7: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A3, Def1;
A8: succ (k + 1) = (k + 1) + 1 by NAT_1:38;
A9: 1 <= k by A3, XXREAL_0:2;
let K be Subset of the Points of (G_ (k,X)); ::_thesis: ( not K is maximal_clique or K is STAR or K is TOP )
A10: succ k = k + 1 by NAT_1:38;
0 c= card K ;
then 0 in succ (card K) by ORDINAL1:22;
then A11: ( card K = 0 or 0 in card K ) by ORDINAL1:8;
assume A12: K is maximal_clique ; ::_thesis: ( K is STAR or K is TOP )
then A13: K is clique by Def3;
A14: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A3, A6, Def1;
k <= k + 1 by NAT_1:11;
then A15: k c= k + 1 by NAT_1:39;
then A16: k c= card X by A6, XBOOLE_1:1;
K <> {}
proof
consider A1 being set such that
A17: A1 c= X and
A18: card A1 = k by A16, CARD_FIL:36;
A1 in the Points of (G_ (k,X)) by A7, A17, A18;
then consider A being POINT of (G_ (k,X)) such that
A19: A = A1 ;
card A <> card X
proof
assume card A = card X ; ::_thesis: contradiction
then k + 1 c= k by A4, A5, A18, A19, XBOOLE_1:1;
then k in k by A10, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
then card A in card X by A16, A18, A19, CARD_1:3;
then X \ A <> {} by CARD_1:68;
then consider x being set such that
A20: x in X \ A by XBOOLE_0:def_1;
{x} c= X by A20, ZFMISC_1:31;
then A21: A \/ {x} c= X by A17, A19, XBOOLE_1:8;
A22: not x in A by A20, XBOOLE_0:def_5;
A is finite by A18, A19;
then card (A \/ {x}) = k + 1 by A18, A19, A22, CARD_2:41;
then A \/ {x} in the Lines of (G_ (k,X)) by A14, A21;
then consider L being LINE of (G_ (k,X)) such that
A23: L = A \/ {x} ;
consider U being Subset of the Points of (G_ (k,X)) such that
A24: U = {A} ;
A c= L by A23, XBOOLE_1:7;
then A25: A on L by A3, A6, Th10;
A26: U is clique
proof
let B, C be POINT of (G_ (k,X)); :: according to COMBGRAS:def_2 ::_thesis: ( B in U & C in U implies ex L being LINE of (G_ (k,X)) st {B,C} on L )
assume ( B in U & C in U ) ; ::_thesis: ex L being LINE of (G_ (k,X)) st {B,C} on L
then ( B on L & C on L ) by A25, A24, TARSKI:def_1;
then {B,C} on L by INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {B,C} on L ; ::_thesis: verum
end;
assume A27: K = {} ; ::_thesis: contradiction
then K c= U by XBOOLE_1:2;
hence contradiction by A12, A27, A24, A26, Def3; ::_thesis: verum
end;
then 1 c= card K by A1, A11, ORDINAL1:21;
then 1 in succ (card K) by ORDINAL1:22;
then A28: ( card K = 1 or 1 in card K ) by ORDINAL1:8;
A29: k - 1 is Element of NAT by A3, NAT_1:21, XXREAL_0:2;
A30: card K <> 1
proof
assume card K = 1 ; ::_thesis: contradiction
then consider A3 being set such that
A31: K = {A3} by CARD_2:42;
A32: A3 in K by A31, TARSKI:def_1;
then consider A being POINT of (G_ (k,X)) such that
A33: A = A3 ;
A3 in the Points of (G_ (k,X)) by A32;
then A34: ex A4 being Subset of X st
( A = A4 & card A4 = k ) by A7, A33;
then A35: A is finite ;
A36: card A <> card X
proof
assume card A = card X ; ::_thesis: contradiction
then k + 1 c= k by A4, A5, A34, XBOOLE_1:1;
then k in k by A10, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
card A c= card X by A6, A15, A34, XBOOLE_1:1;
then card A in card X by A36, CARD_1:3;
then X \ A <> {} by CARD_1:68;
then consider x being set such that
A37: x in X \ A by XBOOLE_0:def_1;
A38: {x} c= X by A37, ZFMISC_1:31;
then A39: A \/ {x} c= X by A34, XBOOLE_1:8;
not x in A by A37, XBOOLE_0:def_5;
then card (A \/ {x}) = k + 1 by A34, A35, CARD_2:41;
then A \/ {x} in the Lines of (G_ (k,X)) by A14, A39;
then consider L being LINE of (G_ (k,X)) such that
A40: L = A \/ {x} ;
k - 1 <= (k - 1) + 1 by A29, NAT_1:11;
then k - 1 c= card A by A29, A34, NAT_1:39;
then consider B2 being set such that
A41: B2 c= A and
A42: card B2 = k - 1 by A29, CARD_FIL:36;
A43: B2 is finite by A3, A42, NAT_1:21, XXREAL_0:2;
B2 c= X by A34, A41, XBOOLE_1:1;
then A44: B2 \/ {x} c= X by A38, XBOOLE_1:8;
not x in B2 by A37, A41, XBOOLE_0:def_5;
then card (B2 \/ {x}) = (k - 1) + 1 by A42, A43, CARD_2:41;
then B2 \/ {x} in the Points of (G_ (k,X)) by A7, A44;
then consider A1 being POINT of (G_ (k,X)) such that
A45: A1 = B2 \/ {x} ;
A46: {x} c= L by A40, XBOOLE_1:7;
A47: A c= L by A40, XBOOLE_1:7;
then B2 c= L by A41, XBOOLE_1:1;
then A1 c= L by A45, A46, XBOOLE_1:8;
then A48: A1 on L by A3, A6, Th10;
{x} c= A1 by A45, XBOOLE_1:7;
then x in A1 by ZFMISC_1:31;
then A49: A <> A1 by A37, XBOOLE_0:def_5;
consider U being Subset of the Points of (G_ (k,X)) such that
A50: U = {A,A1} ;
A51: A on L by A3, A6, A47, Th10;
A52: U is clique
proof
let B1, B2 be POINT of (G_ (k,X)); :: according to COMBGRAS:def_2 ::_thesis: ( B1 in U & B2 in U implies ex L being LINE of (G_ (k,X)) st {B1,B2} on L )
assume ( B1 in U & B2 in U ) ; ::_thesis: ex L being LINE of (G_ (k,X)) st {B1,B2} on L
then ( B1 on L & B2 on L ) by A51, A48, A50, TARSKI:def_2;
then {B1,B2} on L by INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {B1,B2} on L ; ::_thesis: verum
end;
A in U by A50, TARSKI:def_2;
then A53: K c= U by A31, A33, ZFMISC_1:31;
A1 in U by A50, TARSKI:def_2;
then U <> K by A31, A33, A49, TARSKI:def_1;
hence contradiction by A12, A53, A52, Def3; ::_thesis: verum
end;
succ 1 = 1 + 1 ;
then A54: 2 c= card K by A30, A28, ORDINAL1:21;
then consider R being set such that
A55: R c= K and
A56: card R = 2 by CARD_FIL:36;
consider A1, B1 being set such that
A57: A1 <> B1 and
A58: R = {A1,B1} by A56, CARD_2:60;
A59: A1 in R by A58, TARSKI:def_2;
then A60: A1 in the Points of (G_ (k,X)) by A55, TARSKI:def_3;
then consider A being POINT of (G_ (k,X)) such that
A61: A = A1 ;
A62: B1 in R by A58, TARSKI:def_2;
then A63: B1 in the Points of (G_ (k,X)) by A55, TARSKI:def_3;
then consider B being POINT of (G_ (k,X)) such that
A64: B = B1 ;
consider L being LINE of (G_ (k,X)) such that
A65: {A,B} on L by A13, A55, A59, A62, A61, A64, Def2;
L in the Lines of (G_ (k,X)) ;
then A66: ex L1 being Subset of X st
( L1 = L & card L1 = k + 1 ) by A14;
then A67: L is finite ;
A on L by A65, INCSP_1:1;
then A68: A c= L by A3, A6, Th10;
then A69: A /\ B c= L by XBOOLE_1:108;
then A70: A /\ B c= X by A66, XBOOLE_1:1;
B on L by A65, INCSP_1:1;
then A71: B c= L by A3, A6, Th10;
then A72: A \/ B c= L by A68, XBOOLE_1:8;
then A73: card (A \/ B) c= k + 1 by A66, CARD_1:11;
A74: ex B2 being Subset of X st
( B2 = B & card B2 = k ) by A7, A63, A64;
then A75: B is finite ;
A76: ex A2 being Subset of X st
( A2 = A & card A2 = k ) by A7, A60, A61;
then A77: A is finite ;
A78: k + 1 c= card (A \/ B) by A57, A61, A64, A76, A74, Th1;
then A79: card (A \/ B) = k + 1 by A73, XBOOLE_0:def_10;
then A80: A \/ B = L by A68, A71, A66, A67, CARD_FIN:1, XBOOLE_1:8;
A81: ( ( for C being POINT of (G_ (k,X)) holds
( not C in K or not C on L or not A <> C or not B <> C ) ) implies K is STAR )
proof
A82: card L <> card X
proof
assume card L = card X ; ::_thesis: contradiction
then k + 1 in k + 1 by A4, A8, A66, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
card L c= card X by A4, A5, A66, XBOOLE_1:1;
then card L in card X by A82, CARD_1:3;
then X \ L <> {} by CARD_1:68;
then consider x being set such that
A83: x in X \ L by XBOOLE_0:def_1;
A84: ( not x in A & not x in B ) by A68, A71, A83, XBOOLE_0:def_5;
A85: ( A /\ {x} = {} & B /\ {x} = {} )
proof
assume ( A /\ {x} <> {} or B /\ {x} <> {} ) ; ::_thesis: contradiction
then consider z1 being set such that
A86: ( z1 in A /\ {x} or z1 in B /\ {x} ) by XBOOLE_0:def_1;
( ( z1 in A & z1 in {x} ) or ( z1 in B & z1 in {x} ) ) by A86, XBOOLE_0:def_4;
hence contradiction by A84, TARSKI:def_1; ::_thesis: verum
end;
A87: ( card A = (k - 1) + 1 & card (A \/ B) = (k - 1) + (2 * 1) ) by A76, A73, A78, XBOOLE_0:def_10;
then A88: card (A /\ B) = k - 1 by A29, A74, Th2;
then card (A \ (A /\ B)) = k - (k - 1) by A76, A77, CARD_2:44, XBOOLE_1:17;
then consider z1 being set such that
A89: A \ (A /\ B) = {z1} by CARD_2:42;
card (B \ (A /\ B)) = k - (k - 1) by A74, A75, A88, CARD_2:44, XBOOLE_1:17;
then consider z2 being set such that
A90: B \ (A /\ B) = {z2} by CARD_2:42;
A91: B = (A /\ B) \/ {z2} by A90, XBOOLE_1:17, XBOOLE_1:45;
A92: z2 in {z2} by TARSKI:def_1;
A93: card (A \/ B) = (k - 1) + (2 * 1) by A73, A78, XBOOLE_0:def_10;
A94: not x in A /\ B by A69, A83, XBOOLE_0:def_5;
card A = (k - 1) + 1 by A76;
then card (A /\ B) = k - 1 by A29, A74, A93, Th2;
then A95: card ((A /\ B) \/ {x}) = (k - 1) + 1 by A68, A67, A94, CARD_2:41;
{x} c= X by A83, ZFMISC_1:31;
then A96: (A /\ B) \/ {x} c= X by A70, XBOOLE_1:8;
then (A /\ B) \/ {x} in the Points of (G_ (k,X)) by A7, A95;
then consider C being POINT of (G_ (k,X)) such that
A97: C = (A /\ B) \/ {x} ;
A98: B \/ C c= X by A74, A96, A97, XBOOLE_1:8;
A99: A \/ C c= X by A76, A96, A97, XBOOLE_1:8;
A100: 1 + 1 <= k + 1 by A9, XREAL_1:7;
A101: A /\ B c= B by XBOOLE_1:17;
B /\ C = (B /\ {x}) \/ (B /\ (B /\ A)) by A97, XBOOLE_1:23;
then B /\ C = (B /\ {x}) \/ ((B /\ B) /\ A) by XBOOLE_1:16;
then card (B /\ C) = k - 1 by A29, A74, A85, A87, Th2;
then card (B \/ C) = (k - 1) + (2 * 1) by A29, A74, A95, A97, Th2;
then B \/ C in the Lines of (G_ (k,X)) by A14, A98;
then consider L2 being LINE of (G_ (k,X)) such that
A102: L2 = B \/ C ;
A /\ C = (A /\ {x}) \/ (A /\ (A /\ B)) by A97, XBOOLE_1:23;
then A /\ C = (A /\ {x}) \/ ((A /\ A) /\ B) by XBOOLE_1:16;
then card (A /\ C) = k - 1 by A29, A74, A85, A87, Th2;
then card (A \/ C) = (k - 1) + (2 * 1) by A29, A76, A95, A97, Th2;
then A \/ C in the Lines of (G_ (k,X)) by A14, A99;
then consider L1 being LINE of (G_ (k,X)) such that
A103: L1 = A \/ C ;
A104: {A,B,C} is clique
proof
let Z1, Z2 be POINT of (G_ (k,X)); :: according to COMBGRAS:def_2 ::_thesis: ( Z1 in {A,B,C} & Z2 in {A,B,C} implies ex L being LINE of (G_ (k,X)) st {Z1,Z2} on L )
assume that
A105: Z1 in {A,B,C} and
A106: Z2 in {A,B,C} ; ::_thesis: ex L being LINE of (G_ (k,X)) st {Z1,Z2} on L
A107: ( Z2 = A or Z2 = B or Z2 = C ) by A106, ENUMSET1:def_1;
( Z1 = A or Z1 = B or Z1 = C ) by A105, ENUMSET1:def_1;
then ( ( Z1 c= A \/ B & Z2 c= A \/ B ) or ( Z1 c= A \/ C & Z2 c= A \/ C ) or ( Z1 c= B \/ C & Z2 c= B \/ C ) ) by A107, XBOOLE_1:11;
then ( ( Z1 on L & Z2 on L ) or ( Z1 on L1 & Z2 on L1 ) or ( Z1 on L2 & Z2 on L2 ) ) by A3, A6, A80, A103, A102, Th10;
then ( {Z1,Z2} on L or {Z1,Z2} on L1 or {Z1,Z2} on L2 ) by INCSP_1:1;
hence ex L being LINE of (G_ (k,X)) st {Z1,Z2} on L ; ::_thesis: verum
end;
A108: ( C <> A & C <> B )
proof
assume A109: ( A = C or B = C ) ; ::_thesis: contradiction
{x} c= C by A97, XBOOLE_1:11;
hence contradiction by A84, A109, ZFMISC_1:31; ::_thesis: verum
end;
A110: 3 c= card K
proof
assume not 3 c= card K ; ::_thesis: contradiction
then card K in 3 by ORDINAL1:16;
then card K c= 2 by A2, ORDINAL1:22;
then ( card K = 2 & K is finite ) by A54, XBOOLE_0:def_10;
then A111: K = {A,B} by A55, A56, A58, A61, A64, CARD_FIN:1;
( A in {A,B,C} & B in {A,B,C} ) by ENUMSET1:def_1;
then A112: {A,B} c= {A,B,C} by ZFMISC_1:32;
C in {A,B,C} by ENUMSET1:def_1;
then not {A,B,C} c= {A,B} by A108, TARSKI:def_2;
hence contradiction by A12, A104, A111, A112, Def3; ::_thesis: verum
end;
card {A,B} <> card K
proof
assume card {A,B} = card K ; ::_thesis: contradiction
then 3 in 3 by A2, A56, A58, A61, A64, A110, ORDINAL1:22;
hence contradiction ; ::_thesis: verum
end;
then card {A,B} in card K by A54, A56, A58, A61, A64, CARD_1:3;
then K \ {A,B} <> {} by CARD_1:68;
then consider E2 being set such that
A113: E2 in K \ {A,B} by XBOOLE_0:def_1;
A114: card A = (k - 1) + 1 by A76;
then A115: card (A /\ B) = (k + 1) - 2 by A29, A74, A93, Th2;
A116: card B = (k - 1) + 1 by A74;
A117: A /\ B c= A by XBOOLE_1:17;
A118: not E2 in {A,B} by A113, XBOOLE_0:def_5;
then A119: E2 <> A by TARSKI:def_2;
E2 in the Points of (G_ (k,X)) by A113;
then consider E1 being Subset of X such that
A120: E1 = E2 and
A121: card E1 = k by A7;
consider E being POINT of (G_ (k,X)) such that
A122: E = E1 by A113, A120;
A123: A = (A /\ B) \/ {z1} by A89, XBOOLE_1:17, XBOOLE_1:45;
A124: z1 in {z1} by TARSKI:def_1;
then A125: not z1 in A /\ B by A89, XBOOLE_0:def_5;
A126: ( card A = (k + 1) - 1 & 2 + 1 <= k + 1 ) by A3, A76, XREAL_1:7;
consider S being set such that
A127: S = { D where D is Subset of X : ( card D = k & A /\ B c= D ) } ;
A128: E2 in K by A113, XBOOLE_0:def_5;
then consider K1 being LINE of (G_ (k,X)) such that
A129: {A,E} on K1 by A13, A55, A59, A61, A120, A122, Def2;
consider K2 being LINE of (G_ (k,X)) such that
A130: {B,E} on K2 by A13, A55, A62, A64, A128, A120, A122, Def2;
E on K2 by A130, INCSP_1:1;
then A131: E c= K2 by A3, A6, Th10;
K2 in the Lines of (G_ (k,X)) ;
then A132: ex M2 being Subset of X st
( K2 = M2 & card M2 = k + 1 ) by A14;
B on K2 by A130, INCSP_1:1;
then B c= K2 by A3, A6, Th10;
then B \/ E c= K2 by A131, XBOOLE_1:8;
then A133: card (B \/ E) c= k + 1 by A132, CARD_1:11;
A134: E2 <> B by A118, TARSKI:def_2;
then k + 1 c= card (B \/ E) by A74, A120, A121, A122, Th1;
then card (B \/ E) = (k - 1) + (2 * 1) by A133, XBOOLE_0:def_10;
then A135: card (B /\ E) = (k + 1) - 2 by A29, A121, A122, A116, Th2;
assume for C being POINT of (G_ (k,X)) holds
( not C in K or not C on L or not A <> C or not B <> C ) ; ::_thesis: K is STAR
then A136: not E on L by A128, A119, A134, A120, A122;
A137: not card ((A \/ B) \/ E) = k + 1
proof
assume A138: card ((A \/ B) \/ E) = k + 1 ; ::_thesis: contradiction
then ( A \/ B c= (A \/ B) \/ E & (A \/ B) \/ E is finite ) by XBOOLE_1:7;
then A139: A \/ B = (A \/ B) \/ E by A79, A138, CARD_FIN:1;
E c= (A \/ B) \/ E by XBOOLE_1:7;
then E c= L by A72, A139, XBOOLE_1:1;
hence contradiction by A3, A6, A136, Th10; ::_thesis: verum
end;
E on K1 by A129, INCSP_1:1;
then A140: E c= K1 by A3, A6, Th10;
K1 in the Lines of (G_ (k,X)) ;
then A141: ex M1 being Subset of X st
( K1 = M1 & card M1 = k + 1 ) by A14;
A on K1 by A129, INCSP_1:1;
then A c= K1 by A3, A6, Th10;
then A \/ E c= K1 by A140, XBOOLE_1:8;
then A142: card (A \/ E) c= k + 1 by A141, CARD_1:11;
k + 1 c= card (A \/ E) by A76, A119, A120, A121, A122, Th1;
then card (A \/ E) = (k - 1) + (2 * 1) by A142, XBOOLE_0:def_10;
then card (A /\ E) = (k + 1) - 2 by A29, A121, A122, A114, Th2;
then ( ( card ((A /\ B) /\ E) = (k + 1) - 2 & card ((A \/ B) \/ E) = (k + 1) + 1 ) or ( card ((A /\ B) /\ E) = (k + 1) - 3 & card ((A \/ B) \/ E) = k + 1 ) ) by A74, A121, A122, A135, A126, A100, A115, Th7;
then A143: A /\ B = (A /\ B) /\ E by A68, A67, A88, A137, CARD_FIN:1, XBOOLE_1:17;
then A144: A /\ B c= E by XBOOLE_1:17;
E is finite by A121, A122;
then card (E \ (A /\ B)) = k - (k - 1) by A88, A121, A122, A143, CARD_2:44, XBOOLE_1:17;
then consider z4 being set such that
A145: E \ (A /\ B) = {z4} by CARD_2:42;
A146: E = (A /\ B) \/ {z4} by A143, A145, XBOOLE_1:17, XBOOLE_1:45;
A147: K c= S
proof
assume not K c= S ; ::_thesis: contradiction
then consider D2 being set such that
A148: D2 in K and
A149: not D2 in S by TARSKI:def_3;
D2 in the Points of (G_ (k,X)) by A148;
then consider D1 being Subset of X such that
A150: D1 = D2 and
A151: card D1 = k by A7;
consider D being POINT of (G_ (k,X)) such that
A152: D = D1 by A148, A150;
consider K11 being LINE of (G_ (k,X)) such that
A153: {A,D} on K11 by A13, A55, A59, A61, A148, A150, A152, Def2;
D on K11 by A153, INCSP_1:1;
then A154: D c= K11 by A3, A6, Th10;
K11 in the Lines of (G_ (k,X)) ;
then A155: ex R11 being Subset of X st
( R11 = K11 & card R11 = k + 1 ) by A14;
A156: card D = (k - 1) + 1 by A151, A152;
consider K13 being LINE of (G_ (k,X)) such that
A157: {E,D} on K13 by A13, A128, A120, A122, A148, A150, A152, Def2;
consider K12 being LINE of (G_ (k,X)) such that
A158: {B,D} on K12 by A13, A55, A62, A64, A148, A150, A152, Def2;
A on K11 by A153, INCSP_1:1;
then A c= K11 by A3, A6, Th10;
then A \/ D c= K11 by A154, XBOOLE_1:8;
then A159: card (A \/ D) c= k + 1 by A155, CARD_1:11;
A <> D by A127, A117, A149, A150, A151, A152;
then k + 1 c= card (A \/ D) by A76, A151, A152, Th1;
then card (A \/ D) = (k - 1) + (2 * 1) by A159, XBOOLE_0:def_10;
then A160: card (A /\ D) = k - 1 by A29, A76, A156, Th2;
not A /\ B c= D by A127, A149, A150, A151, A152;
then ex y being set st
( y in A /\ B & not y in D ) by TARSKI:def_3;
then A /\ B <> (A /\ B) /\ D by XBOOLE_0:def_4;
then A161: card ((A /\ B) /\ D) <> card (A /\ B) by A77, CARD_FIN:1, XBOOLE_1:17;
D on K13 by A157, INCSP_1:1;
then A162: D c= K13 by A3, A6, Th10;
K13 in the Lines of (G_ (k,X)) ;
then A163: ex R13 being Subset of X st
( R13 = K13 & card R13 = k + 1 ) by A14;
D on K12 by A158, INCSP_1:1;
then A164: D c= K12 by A3, A6, Th10;
K12 in the Lines of (G_ (k,X)) ;
then A165: ex R12 being Subset of X st
( R12 = K12 & card R12 = k + 1 ) by A14;
B on K12 by A158, INCSP_1:1;
then B c= K12 by A3, A6, Th10;
then B \/ D c= K12 by A164, XBOOLE_1:8;
then A166: card (B \/ D) c= k + 1 by A165, CARD_1:11;
B <> D by A127, A101, A149, A150, A151, A152;
then k + 1 c= card (B \/ D) by A74, A151, A152, Th1;
then card (B \/ D) = (k - 1) + (2 * 1) by A166, XBOOLE_0:def_10;
then A167: card (B /\ D) = k - 1 by A29, A74, A156, Th2;
E on K13 by A157, INCSP_1:1;
then E c= K13 by A3, A6, Th10;
then E \/ D c= K13 by A162, XBOOLE_1:8;
then A168: card (E \/ D) c= k + 1 by A163, CARD_1:11;
E <> D by A127, A144, A149, A150, A151, A152;
then k + 1 c= card (E \/ D) by A121, A122, A151, A152, Th1;
then card (E \/ D) = (k - 1) + (2 * 1) by A168, XBOOLE_0:def_10;
then A169: card (E /\ D) = k - 1 by A29, A121, A122, A156, Th2;
A170: ( z1 in D & z2 in D & z4 in D )
proof
assume ( not z1 in D or not z2 in D or not z4 in D ) ; ::_thesis: contradiction
then ( ( A /\ D = ((A /\ B) \/ {z1}) /\ D & {z1} misses D ) or ( B /\ D = ((A /\ B) \/ {z2}) /\ D & {z2} misses D ) or ( E /\ D = ((A /\ B) \/ {z4}) /\ D & {z4} misses D ) ) by A89, A90, A143, A145, XBOOLE_1:17, XBOOLE_1:45, ZFMISC_1:50;
then ( ( A /\ D = ((A /\ B) /\ D) \/ ({z1} /\ D) & {z1} /\ D = {} ) or ( B /\ D = ((A /\ B) /\ D) \/ ({z2} /\ D) & {z2} /\ D = {} ) or ( E /\ D = ((A /\ B) /\ D) \/ ({z4} /\ D) & {z4} /\ D = {} ) ) by XBOOLE_0:def_7, XBOOLE_1:23;
hence contradiction by A29, A74, A87, A160, A167, A169, A161, Th2; ::_thesis: verum
end;
then ( {z1,z2} c= D & {z4} c= D ) by ZFMISC_1:31, ZFMISC_1:32;
then {z1,z2} \/ {z4} c= D by XBOOLE_1:8;
then ( (A /\ B) /\ D c= D & {z1,z2,z4} c= D ) by ENUMSET1:3, XBOOLE_1:17;
then A171: ((A /\ B) /\ D) \/ {z1,z2,z4} c= D by XBOOLE_1:8;
A172: ( z4 in E \ (A /\ B) & (A /\ B) /\ D c= A /\ B ) by A145, TARSKI:def_1, XBOOLE_1:17;
A173: {z1,z2,z4} misses (A /\ B) /\ D
proof
assume not {z1,z2,z4} misses (A /\ B) /\ D ; ::_thesis: contradiction
then {z1,z2,z4} /\ ((A /\ B) /\ D) <> {} by XBOOLE_0:def_7;
then consider m being set such that
A174: m in {z1,z2,z4} /\ ((A /\ B) /\ D) by XBOOLE_0:def_1;
m in {z1,z2,z4} by A174, XBOOLE_0:def_4;
then A175: ( m = z1 or m = z2 or m = z4 ) by ENUMSET1:def_1;
m in (A /\ B) /\ D by A174, XBOOLE_0:def_4;
hence contradiction by A89, A90, A124, A92, A172, A175, XBOOLE_0:def_5; ::_thesis: verum
end;
reconsider r = card ((A /\ B) /\ D) as Nat by A77;
A176: not z1 in (A /\ B) /\ D by A125, XBOOLE_0:def_4;
A /\ D = ((A /\ B) \/ {z1}) /\ D by A89, XBOOLE_1:17, XBOOLE_1:45;
then A /\ D = ((A /\ B) /\ D) \/ ({z1} /\ D) by XBOOLE_1:23;
then A /\ D = ((A /\ B) /\ D) \/ {z1} by A170, ZFMISC_1:46;
then A177: card (A /\ D) = r + 1 by A77, A176, CARD_2:41;
card {z1,z2,z4} = 3 by A57, A61, A64, A123, A91, A119, A134, A120, A122, A146, CARD_2:58;
then card (((A /\ B) /\ D) \/ {z1,z2,z4}) = (k - 2) + 3 by A77, A160, A177, A173, CARD_2:40;
then k + 1 c= k by A151, A152, A171, CARD_1:11;
then k in k by A10, ORDINAL1:21;
hence contradiction ; ::_thesis: verum
end;
S c= the Points of (G_ (k,X))
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in S or Z in the Points of (G_ (k,X)) )
assume Z in S ; ::_thesis: Z in the Points of (G_ (k,X))
then ex Z1 being Subset of X st
( Z = Z1 & card Z1 = k & A /\ B c= Z1 ) by A127;
hence Z in the Points of (G_ (k,X)) by A7; ::_thesis: verum
end;
then consider S1 being Subset of the Points of (G_ (k,X)) such that
A178: S1 = S ;
A179: S1 is STAR by A70, A127, A88, A178, Def4;
then S1 is maximal_clique by A3, A4, Th14;
then S1 is clique by Def3;
hence K is STAR by A12, A147, A178, A179, Def3; ::_thesis: verum
end;
A180: k - 2 is Element of NAT by A3, NAT_1:21;
then A181: succ (k - 2) = (k - 2) + 1 by NAT_1:38;
( ex C being POINT of (G_ (k,X)) st
( C in K & C on L & A <> C & B <> C ) implies K is TOP )
proof
A182: 1 + 1 <= k + 1 by A9, XREAL_1:7;
A183: card B = (k - 1) + 1 by A74;
A184: card A = (k - 1) + 1 by A76;
assume ex C being POINT of (G_ (k,X)) st
( C in K & C on L & A <> C & B <> C ) ; ::_thesis: K is TOP
then consider C being POINT of (G_ (k,X)) such that
A185: C in K and
A186: C on L and
A187: A <> C and
A188: B <> C ;
A189: C c= L by A3, A6, A186, Th10;
then A \/ C c= L by A68, XBOOLE_1:8;
then A190: card (A \/ C) c= k + 1 by A66, CARD_1:11;
B \/ C c= L by A71, A189, XBOOLE_1:8;
then A191: card (B \/ C) c= k + 1 by A66, CARD_1:11;
C in the Points of (G_ (k,X)) ;
then A192: ex C2 being Subset of X st
( C2 = C & card C2 = k ) by A7;
then k + 1 c= card (B \/ C) by A74, A188, Th1;
then card (B \/ C) = (k - 1) + (2 * 1) by A191, XBOOLE_0:def_10;
then A193: card (B /\ C) = (k + 1) - 2 by A29, A192, A183, Th2;
k + 1 c= card (A \/ C) by A76, A187, A192, Th1;
then card (A \/ C) = (k - 1) + (2 * 1) by A190, XBOOLE_0:def_10;
then A194: card (A /\ C) = (k + 1) - 2 by A29, A192, A184, Th2;
A195: card (A \/ B) = (k - 1) + (2 * 1) by A73, A78, XBOOLE_0:def_10;
then A196: A \/ B = L by A68, A71, A66, A67, CARD_FIN:1, XBOOLE_1:8;
A197: A \/ B c= (A \/ B) \/ C by XBOOLE_1:7;
(A \/ B) \/ C c= L by A72, A189, XBOOLE_1:8;
then A198: card ((A \/ B) \/ C) = k + 1 by A195, A196, A197, XBOOLE_0:def_10;
A199: ( card A = (k + 1) - 1 & 2 + 1 <= k + 1 ) by A3, A76, XREAL_1:7;
consider T being set such that
A200: T = { D where D is Subset of X : ( card D = k & D c= L ) } ;
card (A /\ B) = k - 1 by A29, A74, A195, A184, Th2;
then A201: ( ( card ((A /\ B) /\ C) = (k + 1) - 3 & card ((A \/ B) \/ C) = k + 1 ) or ( card ((A /\ B) /\ C) = (k + 1) - 2 & card ((A \/ B) \/ C) = (k + 1) + 1 ) ) by A74, A192, A194, A199, A193, A182, Th7;
A202: K c= T
proof
let D2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not D2 in K or D2 in T )
assume that
A203: D2 in K and
A204: not D2 in T ; ::_thesis: contradiction
D2 in the Points of (G_ (k,X)) by A203;
then consider D1 being Subset of X such that
A205: D1 = D2 and
A206: card D1 = k by A7;
consider D being POINT of (G_ (k,X)) such that
A207: D = D1 by A203, A205;
not D c= L by A200, A204, A205, A206, A207;
then consider x being set such that
A208: x in D and
A209: not x in L by TARSKI:def_3;
A210: card {x} = 1 by CARD_1:30;
A211: D is finite by A206, A207;
A212: card D = (k - 1) + 1 by A206, A207;
{x} c= D by A208, ZFMISC_1:31;
then A213: card (D \ {x}) = k - 1 by A206, A207, A211, A210, CARD_2:44;
consider L13 being LINE of (G_ (k,X)) such that
A214: {C,D} on L13 by A13, A185, A203, A205, A207, Def2;
D on L13 by A214, INCSP_1:1;
then A215: D c= L13 by A3, A6, Th10;
L13 in the Lines of (G_ (k,X)) ;
then A216: ex L23 being Subset of X st
( L23 = L13 & card L23 = k + 1 ) by A14;
C on L13 by A214, INCSP_1:1;
then C c= L13 by A3, A6, Th10;
then C \/ D c= L13 by A215, XBOOLE_1:8;
then A217: card (C \/ D) c= k + 1 by A216, CARD_1:11;
A218: not x in C by A189, A209;
A219: C /\ D c= D \ {x}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in C /\ D or z in D \ {x} )
assume A220: z in C /\ D ; ::_thesis: z in D \ {x}
then z <> x by A218, XBOOLE_0:def_4;
then A221: not z in {x} by TARSKI:def_1;
z in D by A220, XBOOLE_0:def_4;
hence z in D \ {x} by A221, XBOOLE_0:def_5; ::_thesis: verum
end;
C <> D by A189, A200, A204, A205, A206, A207;
then k + 1 c= card (C \/ D) by A192, A206, A207, Th1;
then card (C \/ D) = (k - 1) + (2 * 1) by A217, XBOOLE_0:def_10;
then card (C /\ D) = k - 1 by A29, A192, A212, Th2;
then A222: C /\ D = D \ {x} by A211, A213, A219, CARD_FIN:1;
consider L12 being LINE of (G_ (k,X)) such that
A223: {B,D} on L12 by A13, A55, A62, A64, A203, A205, A207, Def2;
consider L11 being LINE of (G_ (k,X)) such that
A224: {A,D} on L11 by A13, A55, A59, A61, A203, A205, A207, Def2;
D on L11 by A224, INCSP_1:1;
then A225: D c= L11 by A3, A6, Th10;
L11 in the Lines of (G_ (k,X)) ;
then A226: ex L21 being Subset of X st
( L21 = L11 & card L21 = k + 1 ) by A14;
A on L11 by A224, INCSP_1:1;
then A c= L11 by A3, A6, Th10;
then A \/ D c= L11 by A225, XBOOLE_1:8;
then A227: card (A \/ D) c= k + 1 by A226, CARD_1:11;
A228: not x in A by A68, A209;
A229: A /\ D c= D \ {x}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in A /\ D or z in D \ {x} )
assume A230: z in A /\ D ; ::_thesis: z in D \ {x}
then z <> x by A228, XBOOLE_0:def_4;
then A231: not z in {x} by TARSKI:def_1;
z in D by A230, XBOOLE_0:def_4;
hence z in D \ {x} by A231, XBOOLE_0:def_5; ::_thesis: verum
end;
A <> D by A68, A200, A204, A205, A206, A207;
then k + 1 c= card (A \/ D) by A76, A206, A207, Th1;
then A232: card (A \/ D) = (k - 1) + (2 * 1) by A227, XBOOLE_0:def_10;
then card (A /\ D) = k - 1 by A29, A76, A212, Th2;
then A233: A /\ D = D \ {x} by A211, A213, A229, CARD_FIN:1;
D on L12 by A223, INCSP_1:1;
then A234: D c= L12 by A3, A6, Th10;
L12 in the Lines of (G_ (k,X)) ;
then A235: ex L22 being Subset of X st
( L22 = L12 & card L22 = k + 1 ) by A14;
B on L12 by A223, INCSP_1:1;
then B c= L12 by A3, A6, Th10;
then B \/ D c= L12 by A234, XBOOLE_1:8;
then A236: card (B \/ D) c= k + 1 by A235, CARD_1:11;
A237: not x in B by A71, A209;
A238: B /\ D c= D \ {x}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in B /\ D or z in D \ {x} )
assume A239: z in B /\ D ; ::_thesis: z in D \ {x}
then z <> x by A237, XBOOLE_0:def_4;
then A240: not z in {x} by TARSKI:def_1;
z in D by A239, XBOOLE_0:def_4;
hence z in D \ {x} by A240, XBOOLE_0:def_5; ::_thesis: verum
end;
B <> D by A71, A200, A204, A205, A206, A207;
then k + 1 c= card (B \/ D) by A74, A206, A207, Th1;
then card (B \/ D) = (k - 1) + (2 * 1) by A236, XBOOLE_0:def_10;
then card (B /\ D) = k - 1 by A29, A74, A212, Th2;
then B /\ D = D \ {x} by A211, A213, A238, CARD_FIN:1;
then A /\ D = (A /\ D) /\ (B /\ D) by A233;
then A /\ D = (A /\ (D /\ B)) /\ D by XBOOLE_1:16;
then A /\ D = ((A /\ B) /\ D) /\ D by XBOOLE_1:16;
then A /\ D = (A /\ B) /\ (D /\ D) by XBOOLE_1:16;
then A /\ D = ((A /\ B) /\ D) /\ (C /\ D) by A233, A222;
then A /\ D = ((A /\ B) /\ (D /\ C)) /\ D by XBOOLE_1:16;
then A /\ D = (((A /\ B) /\ C) /\ D) /\ D by XBOOLE_1:16;
then A /\ D = ((A /\ B) /\ C) /\ (D /\ D) by XBOOLE_1:16;
then card (((A /\ B) /\ C) /\ D) = k - 1 by A29, A76, A212, A232, Th2;
then k - 1 c= k - 2 by A198, A201, CARD_1:11, XBOOLE_1:17;
then k - 1 in k - 1 by A180, A181, ORDINAL1:22;
hence contradiction ; ::_thesis: verum
end;
T c= the Points of (G_ (k,X))
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in T or e in the Points of (G_ (k,X)) )
assume e in T ; ::_thesis: e in the Points of (G_ (k,X))
then ex E being Subset of X st
( e = E & card E = k & E c= L ) by A200;
hence e in the Points of (G_ (k,X)) by A7; ::_thesis: verum
end;
then consider T1 being Subset of the Points of (G_ (k,X)) such that
A241: T1 = T ;
A242: T1 is TOP by A66, A200, A241, Def5;
then T1 is maximal_clique by A3, A4, Th14;
then T1 is clique by Def3;
hence K is TOP by A12, A202, A241, A242, Def3; ::_thesis: verum
end;
hence ( K is STAR or K is TOP ) by A81; ::_thesis: verum
end;
begin
definition
let S1, S2 be IncProjStr ;
attrc3 is strict ;
struct IncProjMap over S1,S2 -> ;
aggrIncProjMap(# point-map, line-map #) -> IncProjMap over S1,S2;
sel point-map c3 -> Function of the Points of S1, the Points of S2;
sel line-map c3 -> Function of the Lines of S1, the Lines of S2;
end;
definition
let S1, S2 be IncProjStr ;
let F be IncProjMap over S1,S2;
let a be POINT of S1;
funcF . a -> POINT of S2 equals :: COMBGRAS:def 6
the point-map of F . a;
coherence
the point-map of F . a is POINT of S2 ;
end;
:: deftheorem defines . COMBGRAS:def_6_:_
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for a being POINT of S1 holds F . a = the point-map of F . a;
definition
let S1, S2 be IncProjStr ;
let F be IncProjMap over S1,S2;
let L be LINE of S1;
funcF . L -> LINE of S2 equals :: COMBGRAS:def 7
the line-map of F . L;
coherence
the line-map of F . L is LINE of S2 ;
end;
:: deftheorem defines . COMBGRAS:def_7_:_
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for L being LINE of S1 holds F . L = the line-map of F . L;
theorem Th16: :: COMBGRAS:16
for S1, S2 being IncProjStr
for F1, F2 being IncProjMap over S1,S2 st ( for A being POINT of S1 holds F1 . A = F2 . A ) & ( for L being LINE of S1 holds F1 . L = F2 . L ) holds
IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #)
proof
let S1, S2 be IncProjStr ; ::_thesis: for F1, F2 being IncProjMap over S1,S2 st ( for A being POINT of S1 holds F1 . A = F2 . A ) & ( for L being LINE of S1 holds F1 . L = F2 . L ) holds
IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #)
let F1, F2 be IncProjMap over S1,S2; ::_thesis: ( ( for A being POINT of S1 holds F1 . A = F2 . A ) & ( for L being LINE of S1 holds F1 . L = F2 . L ) implies IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) )
assume that
A1: for A being POINT of S1 holds F1 . A = F2 . A and
A2: for L being LINE of S1 holds F1 . L = F2 . L ; ::_thesis: IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #)
for a being set st a in the Points of S1 holds
the point-map of F1 . a = the point-map of F2 . a
proof
let a be set ; ::_thesis: ( a in the Points of S1 implies the point-map of F1 . a = the point-map of F2 . a )
assume a in the Points of S1 ; ::_thesis: the point-map of F1 . a = the point-map of F2 . a
then consider A being POINT of S1 such that
A3: A = a ;
F1 . A = F2 . A by A1;
hence the point-map of F1 . a = the point-map of F2 . a by A3; ::_thesis: verum
end;
then A4: the point-map of F1 = the point-map of F2 by FUNCT_2:12;
for l being set st l in the Lines of S1 holds
the line-map of F1 . l = the line-map of F2 . l
proof
let l be set ; ::_thesis: ( l in the Lines of S1 implies the line-map of F1 . l = the line-map of F2 . l )
assume l in the Lines of S1 ; ::_thesis: the line-map of F1 . l = the line-map of F2 . l
then consider L being LINE of S1 such that
A5: L = l ;
F1 . L = F2 . L by A2;
hence the line-map of F1 . l = the line-map of F2 . l by A5; ::_thesis: verum
end;
hence IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) by A4, FUNCT_2:12; ::_thesis: verum
end;
definition
let S1, S2 be IncProjStr ;
let F be IncProjMap over S1,S2;
attrF is incidence_preserving means :Def8: :: COMBGRAS:def 8
for A1 being POINT of S1
for L1 being LINE of S1 holds
( A1 on L1 iff F . A1 on F . L1 );
end;
:: deftheorem Def8 defines incidence_preserving COMBGRAS:def_8_:_
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2 holds
( F is incidence_preserving iff for A1 being POINT of S1
for L1 being LINE of S1 holds
( A1 on L1 iff F . A1 on F . L1 ) );
theorem :: COMBGRAS:17
for S1, S2 being IncProjStr
for F1, F2 being IncProjMap over S1,S2 st IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) & F1 is incidence_preserving holds
F2 is incidence_preserving
proof
let S1, S2 be IncProjStr ; ::_thesis: for F1, F2 being IncProjMap over S1,S2 st IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) & F1 is incidence_preserving holds
F2 is incidence_preserving
let F1, F2 be IncProjMap over S1,S2; ::_thesis: ( IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) & F1 is incidence_preserving implies F2 is incidence_preserving )
assume that
A1: IncProjMap(# the point-map of F1, the line-map of F1 #) = IncProjMap(# the point-map of F2, the line-map of F2 #) and
A2: F1 is incidence_preserving ; ::_thesis: F2 is incidence_preserving
let A1 be POINT of S1; :: according to COMBGRAS:def_8 ::_thesis: for L1 being LINE of S1 holds
( A1 on L1 iff F2 . A1 on F2 . L1 )
let L1 be LINE of S1; ::_thesis: ( A1 on L1 iff F2 . A1 on F2 . L1 )
( F2 . A1 = F1 . A1 & F2 . L1 = F1 . L1 ) by A1;
hence ( A1 on L1 iff F2 . A1 on F2 . L1 ) by A2, Def8; ::_thesis: verum
end;
definition
let S be IncProjStr ;
let F be IncProjMap over S,S;
attrF is automorphism means :Def9: :: COMBGRAS:def 9
( the line-map of F is bijective & the point-map of F is bijective & F is incidence_preserving );
end;
:: deftheorem Def9 defines automorphism COMBGRAS:def_9_:_
for S being IncProjStr
for F being IncProjMap over S,S holds
( F is automorphism iff ( the line-map of F is bijective & the point-map of F is bijective & F is incidence_preserving ) );
definition
let S1, S2 be IncProjStr ;
let F be IncProjMap over S1,S2;
let K be Subset of the Points of S1;
funcF .: K -> Subset of the Points of S2 equals :: COMBGRAS:def 10
the point-map of F .: K;
coherence
the point-map of F .: K is Subset of the Points of S2
proof
the point-map of F .: K c= the Points of S2
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in the point-map of F .: K or b in the Points of S2 )
assume b in the point-map of F .: K ; ::_thesis: b in the Points of S2
then ex a being set st
( a in dom the point-map of F & a in K & b = the point-map of F . a ) by FUNCT_1:def_6;
hence b in the Points of S2 by FUNCT_2:5; ::_thesis: verum
end;
hence the point-map of F .: K is Subset of the Points of S2 ; ::_thesis: verum
end;
end;
:: deftheorem defines .: COMBGRAS:def_10_:_
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for K being Subset of the Points of S1 holds F .: K = the point-map of F .: K;
definition
let S1, S2 be IncProjStr ;
let F be IncProjMap over S1,S2;
let K be Subset of the Points of S2;
funcF " K -> Subset of the Points of S1 equals :: COMBGRAS:def 11
the point-map of F " K;
coherence
the point-map of F " K is Subset of the Points of S1
proof
the point-map of F " K c= the Points of S1
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in the point-map of F " K or b in the Points of S1 )
assume b in the point-map of F " K ; ::_thesis: b in the Points of S1
then b in dom the point-map of F by FUNCT_1:def_7;
hence b in the Points of S1 ; ::_thesis: verum
end;
hence the point-map of F " K is Subset of the Points of S1 ; ::_thesis: verum
end;
end;
:: deftheorem defines " COMBGRAS:def_11_:_
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for K being Subset of the Points of S2 holds F " K = the point-map of F " K;
definition
let X be set ;
let A be finite set ;
func ^^ (A,X) -> Subset of (bool X) equals :: COMBGRAS:def 12
{ B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } ;
coherence
{ B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } is Subset of (bool X)
proof
set Y = { B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } ;
{ B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } c= bool X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } or x in bool X )
assume x in { B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } ; ::_thesis: x in bool X
then ex B1 being Subset of X st
( x = B1 & card B1 = (card A) + 1 & A c= B1 ) ;
hence x in bool X ; ::_thesis: verum
end;
hence { B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } is Subset of (bool X) ; ::_thesis: verum
end;
end;
:: deftheorem defines ^^ COMBGRAS:def_12_:_
for X being set
for A being finite set holds ^^ (A,X) = { B where B is Subset of X : ( card B = (card A) + 1 & A c= B ) } ;
definition
let k be Element of NAT ;
let X be non empty set ;
assume B1: ( 0 < k & k + 1 c= card X ) ;
let A be finite set ;
assume B2: card A = k - 1 ;
func ^^ (A,X,k) -> Subset of the Points of (G_ (k,X)) equals :Def13: :: COMBGRAS:def 13
^^ (A,X);
coherence
^^ (A,X) is Subset of the Points of (G_ (k,X))
proof
A1: the Points of (G_ (k,X)) = { B where B is Subset of X : card B = k } by B1, Def1;
^^ (A,X) c= the Points of (G_ (k,X))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ^^ (A,X) or x in the Points of (G_ (k,X)) )
assume x in ^^ (A,X) ; ::_thesis: x in the Points of (G_ (k,X))
then ex B1 being Subset of X st
( x = B1 & card B1 = (card A) + 1 & A c= B1 ) ;
hence x in the Points of (G_ (k,X)) by B2, A1; ::_thesis: verum
end;
hence ^^ (A,X) is Subset of the Points of (G_ (k,X)) ; ::_thesis: verum
end;
end;
:: deftheorem Def13 defines ^^ COMBGRAS:def_13_:_
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for A being finite set st card A = k - 1 holds
^^ (A,X,k) = ^^ (A,X);
theorem Th18: :: COMBGRAS:18
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for K being Subset of the Points of S1 holds F .: K = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
proof
let S1, S2 be IncProjStr ; ::_thesis: for F being IncProjMap over S1,S2
for K being Subset of the Points of S1 holds F .: K = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
let F be IncProjMap over S1,S2; ::_thesis: for K being Subset of the Points of S1 holds F .: K = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
let K be Subset of the Points of S1; ::_thesis: F .: K = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
set Image = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } ;
A1: F .: K c= { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in F .: K or b in { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } )
assume b in F .: K ; ::_thesis: b in { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) }
then consider a being set such that
A2: a in dom the point-map of F and
A3: a in K and
A4: b = the point-map of F . a by FUNCT_1:def_6;
consider A being POINT of S1 such that
A5: a = A by A2;
b in the Points of S2 by A2, A4, FUNCT_2:5;
then consider B1 being POINT of S2 such that
A6: b = B1 ;
F . A = B1 by A4, A5, A6;
hence b in { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } by A3, A4, A5; ::_thesis: verum
end;
{ B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } c= F .: K
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } or b in F .: K )
assume b in { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } ; ::_thesis: b in F .: K
then A7: ex B being POINT of S2 st
( B = b & ex A being POINT of S1 st
( A in K & F . A = B ) ) ;
the Points of S1 = dom the point-map of F by FUNCT_2:def_1;
hence b in F .: K by A7, FUNCT_1:def_6; ::_thesis: verum
end;
hence F .: K = { B where B is POINT of S2 : ex A being POINT of S1 st
( A in K & F . A = B ) } by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: COMBGRAS:19
for S1, S2 being IncProjStr
for F being IncProjMap over S1,S2
for K being Subset of the Points of S2 holds F " K = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
proof
let S1, S2 be IncProjStr ; ::_thesis: for F being IncProjMap over S1,S2
for K being Subset of the Points of S2 holds F " K = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
let F be IncProjMap over S1,S2; ::_thesis: for K being Subset of the Points of S2 holds F " K = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
let K be Subset of the Points of S2; ::_thesis: F " K = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
set Image = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } ;
A1: F " K c= { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in F " K or a in { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } )
assume A2: a in F " K ; ::_thesis: a in { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) }
then consider A being POINT of S1 such that
A3: a = A ;
A4: the point-map of F . a in K by A2, FUNCT_1:def_7;
then consider B1 being POINT of S2 such that
A5: the point-map of F . a = B1 ;
F . A = B1 by A3, A5;
hence a in { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } by A4, A3; ::_thesis: verum
end;
{ A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } c= F " K
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } or a in F " K )
assume a in { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } ; ::_thesis: a in F " K
then A6: ex A being POINT of S1 st
( A = a & ex B being POINT of S2 st
( B in K & F . A = B ) ) ;
the Points of S1 = dom the point-map of F by FUNCT_2:def_1;
hence a in F " K by A6, FUNCT_1:def_7; ::_thesis: verum
end;
hence F " K = { A where A is POINT of S1 : ex B being POINT of S2 st
( B in K & F . A = B ) } by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th20: :: COMBGRAS:20
for S being IncProjStr
for F being IncProjMap over S,S
for K being Subset of the Points of S st F is incidence_preserving & K is clique holds
F .: K is clique
proof
let S be IncProjStr ; ::_thesis: for F being IncProjMap over S,S
for K being Subset of the Points of S st F is incidence_preserving & K is clique holds
F .: K is clique
let F be IncProjMap over S,S; ::_thesis: for K being Subset of the Points of S st F is incidence_preserving & K is clique holds
F .: K is clique
let K be Subset of the Points of S; ::_thesis: ( F is incidence_preserving & K is clique implies F .: K is clique )
assume that
A1: F is incidence_preserving and
A2: K is clique ; ::_thesis: F .: K is clique
let B1, B2 be POINT of S; :: according to COMBGRAS:def_2 ::_thesis: ( B1 in F .: K & B2 in F .: K implies ex L being LINE of S st {B1,B2} on L )
assume that
A3: B1 in F .: K and
A4: B2 in F .: K ; ::_thesis: ex L being LINE of S st {B1,B2} on L
A5: F .: K = { B where B is POINT of S : ex A being POINT of S st
( A in K & F . A = B ) } by Th18;
then consider B11 being POINT of S such that
A6: B1 = B11 and
A7: ex A being POINT of S st
( A in K & F . A = B11 ) by A3;
consider B12 being POINT of S such that
A8: B2 = B12 and
A9: ex A being POINT of S st
( A in K & F . A = B12 ) by A5, A4;
consider A12 being POINT of S such that
A10: A12 in K and
A11: F . A12 = B12 by A9;
consider A11 being POINT of S such that
A12: A11 in K and
A13: F . A11 = B11 by A7;
consider L1 being LINE of S such that
A14: {A11,A12} on L1 by A2, A12, A10, Def2;
A12 on L1 by A14, INCSP_1:1;
then A15: F . A12 on F . L1 by A1, Def8;
A11 on L1 by A14, INCSP_1:1;
then F . A11 on F . L1 by A1, Def8;
then {B1,B2} on F . L1 by A6, A8, A13, A11, A15, INCSP_1:1;
hence ex L being LINE of S st {B1,B2} on L ; ::_thesis: verum
end;
theorem Th21: :: COMBGRAS:21
for S being IncProjStr
for F being IncProjMap over S,S
for K being Subset of the Points of S st F is incidence_preserving & the line-map of F is onto & K is clique holds
F " K is clique
proof
let S be IncProjStr ; ::_thesis: for F being IncProjMap over S,S
for K being Subset of the Points of S st F is incidence_preserving & the line-map of F is onto & K is clique holds
F " K is clique
let F be IncProjMap over S,S; ::_thesis: for K being Subset of the Points of S st F is incidence_preserving & the line-map of F is onto & K is clique holds
F " K is clique
let K be Subset of the Points of S; ::_thesis: ( F is incidence_preserving & the line-map of F is onto & K is clique implies F " K is clique )
assume that
A1: F is incidence_preserving and
A2: the line-map of F is onto and
A3: K is clique ; ::_thesis: F " K is clique
let A1, A2 be POINT of S; :: according to COMBGRAS:def_2 ::_thesis: ( A1 in F " K & A2 in F " K implies ex L being LINE of S st {A1,A2} on L )
assume ( A1 in F " K & A2 in F " K ) ; ::_thesis: ex L being LINE of S st {A1,A2} on L
then ( F . A1 in K & F . A2 in K ) by FUNCT_1:def_7;
then consider L2 being LINE of S such that
A4: {(F . A1),(F . A2)} on L2 by A3, Def2;
the Lines of S = rng the line-map of F by A2, FUNCT_2:def_3;
then consider l1 being set such that
A5: l1 in dom the line-map of F and
A6: L2 = the line-map of F . l1 by FUNCT_1:def_3;
consider L1 being LINE of S such that
A7: L1 = l1 by A5;
A8: L2 = F . L1 by A6, A7;
F . A2 on L2 by A4, INCSP_1:1;
then A9: A2 on L1 by A1, A8, Def8;
F . A1 on L2 by A4, INCSP_1:1;
then A1 on L1 by A1, A8, Def8;
then {A1,A2} on L1 by A9, INCSP_1:1;
hence ex L being LINE of S st {A1,A2} on L ; ::_thesis: verum
end;
theorem Th22: :: COMBGRAS:22
for S being IncProjStr
for F being IncProjMap over S,S
for K being Subset of the Points of S st F is automorphism & K is maximal_clique holds
( F .: K is maximal_clique & F " K is maximal_clique )
proof
let S be IncProjStr ; ::_thesis: for F being IncProjMap over S,S
for K being Subset of the Points of S st F is automorphism & K is maximal_clique holds
( F .: K is maximal_clique & F " K is maximal_clique )
let F be IncProjMap over S,S; ::_thesis: for K being Subset of the Points of S st F is automorphism & K is maximal_clique holds
( F .: K is maximal_clique & F " K is maximal_clique )
let K be Subset of the Points of S; ::_thesis: ( F is automorphism & K is maximal_clique implies ( F .: K is maximal_clique & F " K is maximal_clique ) )
assume that
A1: F is automorphism and
A2: K is maximal_clique ; ::_thesis: ( F .: K is maximal_clique & F " K is maximal_clique )
A3: F is incidence_preserving by A1, Def9;
the point-map of F is bijective by A1, Def9;
then A4: the Points of S = rng the point-map of F by FUNCT_2:def_3;
A5: the Points of S = dom the point-map of F by FUNCT_2:52;
A6: for U being Subset of the Points of S st U is clique & F " K c= U holds
U = F " K
proof
let U be Subset of the Points of S; ::_thesis: ( U is clique & F " K c= U implies U = F " K )
assume that
A7: U is clique and
A8: F " K c= U ; ::_thesis: U = F " K
F .: (F " K) c= F .: U by A8, RELAT_1:123;
then A9: K c= F .: U by A4, FUNCT_1:77;
A10: U c= F " (F .: U) by A5, FUNCT_1:76;
F .: U is clique by A3, A7, Th20;
then U c= F " K by A2, A9, A10, Def3;
hence U = F " K by A8, XBOOLE_0:def_10; ::_thesis: verum
end;
A11: the line-map of F is bijective by A1, Def9;
A12: for U being Subset of the Points of S st U is clique & F .: K c= U holds
U = F .: K
proof
A13: K c= F " (F .: K) by A5, FUNCT_1:76;
let U be Subset of the Points of S; ::_thesis: ( U is clique & F .: K c= U implies U = F .: K )
assume that
A14: U is clique and
A15: F .: K c= U ; ::_thesis: U = F .: K
F " (F .: K) c= F " U by A15, RELAT_1:143;
then A16: K c= F " U by A13, XBOOLE_1:1;
F " U is clique by A11, A3, A14, Th21;
then F .: (F " U) c= F .: K by A2, A16, Def3;
then U c= F .: K by A4, FUNCT_1:77;
hence U = F .: K by A15, XBOOLE_0:def_10; ::_thesis: verum
end;
A17: K is clique by A2, Def3;
then A18: F .: K is clique by A3, Th20;
F " K is clique by A11, A17, A3, Th21;
hence ( F .: K is maximal_clique & F " K is maximal_clique ) by A18, A12, A6, Def3; ::_thesis: verum
end;
theorem Th23: :: COMBGRAS:23
for k being Element of NAT
for X being non empty set st 2 <= k & k + 2 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 2 <= k & k + 2 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR )
let X be non empty set ; ::_thesis: ( 2 <= k & k + 2 c= card X implies for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR ) )
assume that
A1: 2 <= k and
A2: k + 2 c= card X ; ::_thesis: for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR )
let F be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( F is automorphism implies for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR ) )
assume A3: F is automorphism ; ::_thesis: for K being Subset of the Points of (G_ (k,X)) st K is STAR holds
( F .: K is STAR & F " K is STAR )
A4: k - 1 is Element of NAT by A1, NAT_1:21, XXREAL_0:2;
then A5: succ (k - 1) = (k - 1) + 1 by NAT_1:38;
2 - 1 <= k - 1 by A1, XREAL_1:9;
then A6: 1 c= k - 1 by A4, NAT_1:39;
A7: 1 <= k by A1, XXREAL_0:2;
then A8: 1 c= k by NAT_1:39;
let K be Subset of the Points of (G_ (k,X)); ::_thesis: ( K is STAR implies ( F .: K is STAR & F " K is STAR ) )
assume A9: K is STAR ; ::_thesis: ( F .: K is STAR & F " K is STAR )
then A10: K is maximal_clique by A1, A2, Th14;
then A11: K is clique by Def3;
k + 1 <= k + 2 by XREAL_1:7;
then k + 1 c= k + 2 by NAT_1:39;
then A12: k + 1 c= card X by A2, XBOOLE_1:1;
then A13: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, Def1;
A14: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A12, Def1;
A15: k + 0 <= k + 1 by XREAL_1:7;
then 1 <= k + 1 by A7, XXREAL_0:2;
then A16: 1 c= k + 1 by NAT_1:39;
A17: not F " K is TOP
proof
assume F " K is TOP ; ::_thesis: contradiction
then consider B being Subset of X such that
A18: ( card B = k + 1 & F " K = { A where A is Subset of X : ( card A = k & A c= B ) } ) by Def5;
consider X1 being set such that
A19: ( X1 c= B & card X1 = 1 ) by A16, A18, CARD_FIL:36;
A20: B is finite by A18;
then A21: card (B \ X1) = (k + 1) - 1 by A18, A19, CARD_2:44;
then consider X2 being set such that
A22: X2 c= B \ X1 and
A23: card X2 = 1 by A8, CARD_FIL:36;
consider m being Nat such that
A24: m = k - 1 by A4;
A25: card (B \ X2) = (k + 1) - 1 by A18, A20, A22, A23, CARD_2:44, XBOOLE_1:106;
then B \ X2 in the Points of (G_ (k,X)) by A13;
then consider B2 being POINT of (G_ (k,X)) such that
A26: B \ X2 = B2 ;
card ((B \ X1) \ X2) = k - 1 by A20, A21, A22, A23, CARD_2:44;
then consider X3 being set such that
A27: X3 c= (B \ X1) \ X2 and
A28: card X3 = 1 by A6, CARD_FIL:36;
A29: X3 c= B \ (X2 \/ X1) by A27, XBOOLE_1:41;
then A30: card (B \ X3) = (k + 1) - 1 by A18, A20, A28, CARD_2:44, XBOOLE_1:106;
then B \ X3 in the Points of (G_ (k,X)) by A13;
then consider B3 being POINT of (G_ (k,X)) such that
A31: B \ X3 = B3 ;
B in the Lines of (G_ (k,X)) by A14, A18;
then consider L2 being LINE of (G_ (k,X)) such that
A32: B = L2 ;
B \ X1 in the Points of (G_ (k,X)) by A13, A21;
then consider B1 being POINT of (G_ (k,X)) such that
A33: B \ X1 = B1 ;
consider S being Subset of X such that
A34: card S = k - 1 and
A35: K = { A where A is Subset of X : ( card A = k & S c= A ) } by A9, Def4;
consider A1 being POINT of (G_ (k,X)) such that
A36: A1 = F . B1 ;
A37: X3 c= (B \ X2) \ X1 by A29, XBOOLE_1:41;
A38: ( B \ X1 <> B \ X2 & B \ X2 <> B \ X3 & B \ X1 <> B \ X3 )
proof
assume ( B \ X1 = B \ X2 or B \ X2 = B \ X3 or B \ X1 = B \ X3 ) ; ::_thesis: contradiction
then ( X2 = {} or X3 = {} or X3 = {} ) by A22, A27, A37, XBOOLE_1:38, XBOOLE_1:106;
hence contradiction by A23, A28; ::_thesis: verum
end;
consider A3 being POINT of (G_ (k,X)) such that
A39: A3 = F . B3 ;
A40: B \ X3 c= B by XBOOLE_1:106;
then B3 in F " K by A18, A30, A31;
then A41: A3 in K by A39, FUNCT_1:def_7;
then A42: ex A13 being Subset of X st
( A3 = A13 & card A13 = k & S c= A13 ) by A35;
A43: B \ X1 c= B by XBOOLE_1:106;
then B1 in F " K by A18, A21, A33;
then A44: A1 in K by A36, FUNCT_1:def_7;
then A45: ex A11 being Subset of X st
( A1 = A11 & card A11 = k & S c= A11 ) by A35;
then A46: card A1 = (k - 1) + 1 ;
consider A2 being POINT of (G_ (k,X)) such that
A47: A2 = F . B2 ;
A48: B \ X2 c= B by XBOOLE_1:106;
then B2 in F " K by A18, A25, A26;
then A49: A2 in K by A47, FUNCT_1:def_7;
then consider L3a being LINE of (G_ (k,X)) such that
A50: {A1,A2} on L3a by A11, A44, Def2;
A51: card A1 = (k + 1) - 1 by A45;
A52: F is incidence_preserving by A3, Def9;
A53: card ((A1 /\ A2) /\ A3) c= card (A1 /\ A2) by CARD_1:11, XBOOLE_1:17;
consider L1 being LINE of (G_ (k,X)) such that
A54: L1 = F . L2 ;
B1 on L2 by A1, A12, A43, A33, A32, Th10;
then A55: A1 on L1 by A52, A36, A54, Def8;
then A56: A1 c= L1 by A1, A12, Th10;
L1 in the Lines of (G_ (k,X)) ;
then A57: ex l12 being Subset of X st
( L1 = l12 & card l12 = k + 1 ) by A14;
B3 on L2 by A1, A12, A40, A31, A32, Th10;
then A58: A3 on L1 by A52, A39, A54, Def8;
then A59: A3 c= L1 by A1, A12, Th10;
then A1 \/ A3 c= L1 by A56, XBOOLE_1:8;
then A60: card (A1 \/ A3) c= k + 1 by A57, CARD_1:11;
A61: ex A12 being Subset of X st
( A2 = A12 & card A12 = k & S c= A12 ) by A49, A35;
then A62: card A2 = (k - 1) + 1 ;
B2 on L2 by A1, A12, A48, A26, A32, Th10;
then A63: A2 on L1 by A52, A47, A54, Def8;
then A64: A2 c= L1 by A1, A12, Th10;
then A1 \/ A2 c= L1 by A56, XBOOLE_1:8;
then A65: card (A1 \/ A2) c= k + 1 by A57, CARD_1:11;
A66: ( the point-map of F is bijective & the Points of (G_ (k,X)) = dom the point-map of F ) by A3, Def9, FUNCT_2:52;
then A67: A1 <> A2 by A38, A33, A26, A36, A47, FUNCT_1:def_4;
then k + 1 c= card (A1 \/ A2) by A45, A61, Th1;
then card (A1 \/ A2) = (k - 1) + (2 * 1) by A65, XBOOLE_0:def_10;
then A68: card (A1 /\ A2) = (k + 1) - 2 by A4, A61, A46, Th2;
{A1,A2} on L1 by A55, A63, INCSP_1:1;
then A69: L1 = L3a by A67, A50, INCSP_1:def_10;
consider L3b being LINE of (G_ (k,X)) such that
A70: {A2,A3} on L3b by A11, A49, A41, Def2;
A1 <> A3 by A66, A38, A33, A31, A36, A39, FUNCT_1:def_4;
then k + 1 c= card (A1 \/ A3) by A45, A42, Th1;
then card (A1 \/ A3) = (k - 1) + (2 * 1) by A60, XBOOLE_0:def_10;
then A71: card (A1 /\ A3) = (k + 1) - 2 by A4, A42, A46, Th2;
A3 on L3b by A70, INCSP_1:1;
then A72: A3 c= L3b by A1, A12, Th10;
A2 on L3b by A70, INCSP_1:1;
then A73: A2 c= L3b by A1, A12, Th10;
L3b in the Lines of (G_ (k,X)) ;
then A74: ex l13b being Subset of X st
( L3b = l13b & card l13b = k + 1 ) by A14;
card (A1 /\ A2) in succ (k - 1) by A5, A67, A45, A61, Th1;
then card (A1 /\ A2) c= m by A24, ORDINAL1:22;
then A75: card ((A1 /\ A2) /\ A3) c= m by A53, XBOOLE_1:1;
S c= A1 /\ A2 by A45, A61, XBOOLE_1:19;
then S c= (A1 /\ A2) /\ A3 by A42, XBOOLE_1:19;
then m c= card ((A1 /\ A2) /\ A3) by A34, A24, CARD_1:11;
then A76: k - 1 = card ((A1 /\ A2) /\ A3) by A24, A75, XBOOLE_0:def_10;
A1 on L3a by A50, INCSP_1:1;
then A77: A1 c= L3a by A1, A12, Th10;
A78: k - 1 <> (k + 1) - 3 ;
A2 \/ A3 c= L1 by A64, A59, XBOOLE_1:8;
then A79: card (A2 \/ A3) c= k + 1 by A57, CARD_1:11;
A80: A2 <> A3 by A66, A38, A26, A31, A47, A39, FUNCT_1:def_4;
then k + 1 c= card (A2 \/ A3) by A61, A42, Th1;
then card (A2 \/ A3) = (k - 1) + (2 * 1) by A79, XBOOLE_0:def_10;
then A81: card (A2 /\ A3) = (k + 1) - 2 by A4, A42, A62, Th2;
( 2 + 1 <= k + 1 & 2 <= k + 1 ) by A1, A15, XREAL_1:6, XXREAL_0:2;
then A82: card ((A1 \/ A2) \/ A3) = (k + 1) + 1 by A61, A42, A76, A68, A51, A81, A71, A78, Th7;
A83: L3a <> L3b
proof
assume L3a = L3b ; ::_thesis: contradiction
then A1 \/ A2 c= L3b by A77, A73, XBOOLE_1:8;
then (A1 \/ A2) \/ A3 c= L3b by A72, XBOOLE_1:8;
then k + 2 c= k + 1 by A82, A74, CARD_1:11;
then k + 2 <= k + 1 by NAT_1:39;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
{A2,A3} on L1 by A63, A58, INCSP_1:1;
hence contradiction by A80, A70, A83, A69, INCSP_1:def_10; ::_thesis: verum
end;
A84: not F .: K is TOP
proof
A85: k - 1 <> (k + 1) - 3 ;
assume F .: K is TOP ; ::_thesis: contradiction
then consider B being Subset of X such that
A86: ( card B = k + 1 & F .: K = { A where A is Subset of X : ( card A = k & A c= B ) } ) by Def5;
B in the Lines of (G_ (k,X)) by A14, A86;
then consider L2 being LINE of (G_ (k,X)) such that
A87: B = L2 ;
the line-map of F is bijective by A3, Def9;
then the Lines of (G_ (k,X)) = rng the line-map of F by FUNCT_2:def_3;
then consider l1 being set such that
A88: l1 in dom the line-map of F and
A89: L2 = the line-map of F . l1 by FUNCT_1:def_3;
consider L1 being LINE of (G_ (k,X)) such that
A90: l1 = L1 by A88;
A91: L2 = F . L1 by A89, A90;
consider X1 being set such that
A92: ( X1 c= B & card X1 = 1 ) by A16, A86, CARD_FIL:36;
A93: B is finite by A86;
then A94: card (B \ X1) = (k + 1) - 1 by A86, A92, CARD_2:44;
then consider X2 being set such that
A95: X2 c= B \ X1 and
A96: card X2 = 1 by A8, CARD_FIL:36;
consider m being Nat such that
A97: m = k - 1 by A4;
card ((B \ X1) \ X2) = k - 1 by A93, A94, A95, A96, CARD_2:44;
then consider X3 being set such that
A98: X3 c= (B \ X1) \ X2 and
A99: card X3 = 1 by A6, CARD_FIL:36;
A100: X3 c= B \ (X2 \/ X1) by A98, XBOOLE_1:41;
then A101: card (B \ X3) = (k + 1) - 1 by A86, A93, A99, CARD_2:44, XBOOLE_1:106;
then B \ X3 in the Points of (G_ (k,X)) by A13;
then consider B3 being POINT of (G_ (k,X)) such that
A102: B \ X3 = B3 ;
L1 in the Lines of (G_ (k,X)) ;
then A103: ex l12 being Subset of X st
( L1 = l12 & card l12 = k + 1 ) by A14;
B \ X1 in the Points of (G_ (k,X)) by A13, A94;
then consider B1 being POINT of (G_ (k,X)) such that
A104: B \ X1 = B1 ;
A105: B \ X1 c= B by XBOOLE_1:106;
then A106: B1 on L2 by A1, A12, A104, A87, Th10;
consider S being Subset of X such that
A107: card S = k - 1 and
A108: K = { A where A is Subset of X : ( card A = k & S c= A ) } by A9, Def4;
A109: F is incidence_preserving by A3, Def9;
A110: B \ X3 c= B by XBOOLE_1:106;
then A111: B3 on L2 by A1, A12, A102, A87, Th10;
A112: the point-map of F is bijective by A3, Def9;
then A113: the Points of (G_ (k,X)) = rng the point-map of F by FUNCT_2:def_3;
then consider a3 being set such that
A114: a3 in dom the point-map of F and
A115: B3 = the point-map of F . a3 by FUNCT_1:def_3;
consider A3 being POINT of (G_ (k,X)) such that
A116: a3 = A3 by A114;
consider a1 being set such that
A117: a1 in dom the point-map of F and
A118: B1 = the point-map of F . a1 by A113, FUNCT_1:def_3;
consider A1 being POINT of (G_ (k,X)) such that
A119: a1 = A1 by A117;
B3 in F .: K by A86, A101, A110, A102;
then ex C3 being set st
( C3 in dom the point-map of F & C3 in K & B3 = the point-map of F . C3 ) by FUNCT_1:def_6;
then A120: A3 in K by A112, A114, A115, A116, FUNCT_1:def_4;
then A121: ex A13 being Subset of X st
( A3 = A13 & card A13 = k & S c= A13 ) by A108;
B1 in F .: K by A86, A94, A105, A104;
then ex C1 being set st
( C1 in dom the point-map of F & C1 in K & B1 = the point-map of F . C1 ) by FUNCT_1:def_6;
then A122: A1 in K by A112, A117, A118, A119, FUNCT_1:def_4;
then A123: ex A11 being Subset of X st
( A1 = A11 & card A11 = k & S c= A11 ) by A108;
then A124: card A1 = (k - 1) + 1 ;
A125: B1 = F . A1 by A118, A119;
then A1 on L1 by A109, A106, A91, Def8;
then A126: A1 c= L1 by A1, A12, Th10;
A127: B3 = F . A3 by A115, A116;
then A3 on L1 by A109, A111, A91, Def8;
then A128: A3 c= L1 by A1, A12, Th10;
then A1 \/ A3 c= L1 by A126, XBOOLE_1:8;
then A129: card (A1 \/ A3) c= k + 1 by A103, CARD_1:11;
A130: X3 c= (B \ X2) \ X1 by A100, XBOOLE_1:41;
A131: ( B \ X1 <> B \ X2 & B \ X2 <> B \ X3 & B \ X1 <> B \ X3 )
proof
assume ( B \ X1 = B \ X2 or B \ X2 = B \ X3 or B \ X1 = B \ X3 ) ; ::_thesis: contradiction
then ( X2 = {} or X3 = {} or X3 = {} ) by A95, A98, A130, XBOOLE_1:38, XBOOLE_1:106;
hence contradiction by A96, A99; ::_thesis: verum
end;
then k + 1 c= card (A1 \/ A3) by A104, A102, A118, A115, A119, A116, A123, A121, Th1;
then card (A1 \/ A3) = (k - 1) + (2 * 1) by A129, XBOOLE_0:def_10;
then A132: card (A1 /\ A3) = (k + 1) - 2 by A4, A121, A124, Th2;
A133: card (B \ X2) = (k + 1) - 1 by A86, A93, A95, A96, CARD_2:44, XBOOLE_1:106;
then B \ X2 in the Points of (G_ (k,X)) by A13;
then consider B2 being POINT of (G_ (k,X)) such that
A134: B \ X2 = B2 ;
A135: B \ X2 c= B by XBOOLE_1:106;
then A136: B2 on L2 by A1, A12, A134, A87, Th10;
consider a2 being set such that
A137: a2 in dom the point-map of F and
A138: B2 = the point-map of F . a2 by A113, FUNCT_1:def_3;
consider A2 being POINT of (G_ (k,X)) such that
A139: a2 = A2 by A137;
B2 in F .: K by A86, A133, A135, A134;
then ex C2 being set st
( C2 in dom the point-map of F & C2 in K & B2 = the point-map of F . C2 ) by FUNCT_1:def_6;
then A140: A2 in K by A112, A137, A138, A139, FUNCT_1:def_4;
then A141: ex A12 being Subset of X st
( A2 = A12 & card A12 = k & S c= A12 ) by A108;
then A142: card A2 = (k - 1) + 1 ;
A143: B2 = F . A2 by A138, A139;
then A2 on L1 by A109, A136, A91, Def8;
then A144: A2 c= L1 by A1, A12, Th10;
then A1 \/ A2 c= L1 by A126, XBOOLE_1:8;
then A145: card (A1 \/ A2) c= k + 1 by A103, CARD_1:11;
k + 1 c= card (A1 \/ A2) by A131, A104, A134, A118, A138, A119, A139, A123, A141, Th1;
then card (A1 \/ A2) = (k - 1) + (2 * 1) by A145, XBOOLE_0:def_10;
then A146: card (A1 /\ A2) = (k + 1) - 2 by A4, A141, A124, Th2;
A147: A2 on L1 by A109, A136, A143, A91, Def8;
A2 \/ A3 c= L1 by A144, A128, XBOOLE_1:8;
then A148: card (A2 \/ A3) c= k + 1 by A103, CARD_1:11;
k + 1 c= card (A2 \/ A3) by A131, A134, A102, A138, A115, A139, A116, A141, A121, Th1;
then card (A2 \/ A3) = (k - 1) + (2 * 1) by A148, XBOOLE_0:def_10;
then A149: card (A2 /\ A3) = (k + 1) - 2 by A4, A121, A142, Th2;
A150: card A1 = (k + 1) - 1 by A123;
consider L3a being LINE of (G_ (k,X)) such that
A151: {A1,A2} on L3a by A11, A122, A140, Def2;
card (A1 /\ A2) in k by A131, A104, A134, A118, A138, A119, A139, A123, A141, Th1;
then A152: card (A1 /\ A2) c= m by A5, A97, ORDINAL1:22;
card ((A1 /\ A2) /\ A3) c= card (A1 /\ A2) by CARD_1:11, XBOOLE_1:17;
then A153: card ((A1 /\ A2) /\ A3) c= m by A152, XBOOLE_1:1;
S c= A1 /\ A2 by A123, A141, XBOOLE_1:19;
then S c= (A1 /\ A2) /\ A3 by A121, XBOOLE_1:19;
then m c= card ((A1 /\ A2) /\ A3) by A107, A97, CARD_1:11;
then A154: k - 1 = card ((A1 /\ A2) /\ A3) by A97, A153, XBOOLE_0:def_10;
A1 on L3a by A151, INCSP_1:1;
then A155: A1 c= L3a by A1, A12, Th10;
consider L3b being LINE of (G_ (k,X)) such that
A156: {A2,A3} on L3b by A11, A140, A120, Def2;
A3 on L3b by A156, INCSP_1:1;
then A157: A3 c= L3b by A1, A12, Th10;
A2 on L3b by A156, INCSP_1:1;
then A158: A2 c= L3b by A1, A12, Th10;
L3b in the Lines of (G_ (k,X)) ;
then A159: ex l13b being Subset of X st
( L3b = l13b & card l13b = k + 1 ) by A14;
( 2 + 1 <= k + 1 & 2 <= k + 1 ) by A1, A15, XREAL_1:6, XXREAL_0:2;
then A160: card ((A1 \/ A2) \/ A3) = (k + 1) + 1 by A141, A121, A154, A146, A150, A149, A132, A85, Th7;
A161: L3a <> L3b
proof
assume L3a = L3b ; ::_thesis: contradiction
then A1 \/ A2 c= L3b by A155, A158, XBOOLE_1:8;
then (A1 \/ A2) \/ A3 c= L3b by A157, XBOOLE_1:8;
then k + 2 c= k + 1 by A160, A159, CARD_1:11;
then k + 2 <= k + 1 by NAT_1:39;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
A1 on L1 by A109, A106, A125, A91, Def8;
then {A1,A2} on L1 by A147, INCSP_1:1;
then A162: L1 = L3a by A131, A104, A134, A118, A138, A119, A139, A151, INCSP_1:def_10;
A3 on L1 by A109, A111, A127, A91, Def8;
then {A2,A3} on L1 by A147, INCSP_1:1;
hence contradiction by A131, A134, A102, A138, A115, A139, A116, A156, A161, A162, INCSP_1:def_10; ::_thesis: verum
end;
( F .: K is maximal_clique & F " K is maximal_clique ) by A3, A10, Th22;
hence ( F .: K is STAR & F " K is STAR ) by A1, A2, A84, A17, Th15; ::_thesis: verum
end;
definition
let k be Element of NAT ;
let X be non empty set ;
assume B1: ( 0 < k & k + 1 c= card X ) ;
let s be Permutation of X;
func incprojmap (k,s) -> strict IncProjMap over G_ (k,X), G_ (k,X) means :Def14: :: COMBGRAS:def 14
( ( for A being POINT of (G_ (k,X)) holds it . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds it . L = s .: L ) );
existence
ex b1 being strict IncProjMap over G_ (k,X), G_ (k,X) st
( ( for A being POINT of (G_ (k,X)) holds b1 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds b1 . L = s .: L ) )
proof
deffunc H1( set ) -> set = s .: $1;
consider P being Function such that
A1: ( dom P = the Points of (G_ (k,X)) & ( for x being set st x in the Points of (G_ (k,X)) holds
P . x = H1(x) ) ) from FUNCT_1:sch_3();
A2: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by B1, Def1;
rng P c= the Points of (G_ (k,X))
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng P or b in the Points of (G_ (k,X)) )
assume b in rng P ; ::_thesis: b in the Points of (G_ (k,X))
then consider a being set such that
A3: a in the Points of (G_ (k,X)) and
A4: b = P . a by A1, FUNCT_1:def_3;
consider A being Subset of X such that
A5: A = a and
A6: card A = k by A2, A3;
A7: b = s .: A by A1, A3, A4, A5;
A8: b c= X
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in b or y in X )
assume y in b ; ::_thesis: y in X
then ex x being set st
( x in dom s & x in A & y = s . x ) by A7, FUNCT_1:def_6;
then y in rng s by FUNCT_1:3;
hence y in X by FUNCT_2:def_3; ::_thesis: verum
end;
dom s = X by FUNCT_2:def_1;
then card b = k by A6, A7, Th4;
hence b in the Points of (G_ (k,X)) by A2, A8; ::_thesis: verum
end;
then reconsider P = P as Function of the Points of (G_ (k,X)), the Points of (G_ (k,X)) by A1, FUNCT_2:2;
A9: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by B1, Def1;
consider L being Function such that
A10: ( dom L = the Lines of (G_ (k,X)) & ( for x being set st x in the Lines of (G_ (k,X)) holds
L . x = H1(x) ) ) from FUNCT_1:sch_3();
rng L c= the Lines of (G_ (k,X))
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng L or b in the Lines of (G_ (k,X)) )
assume b in rng L ; ::_thesis: b in the Lines of (G_ (k,X))
then consider a being set such that
A11: a in the Lines of (G_ (k,X)) and
A12: b = L . a by A10, FUNCT_1:def_3;
consider A being Subset of X such that
A13: A = a and
A14: card A = k + 1 by A9, A11;
A15: b = s .: A by A10, A11, A12, A13;
A16: b c= X
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in b or y in X )
assume y in b ; ::_thesis: y in X
then ex x being set st
( x in dom s & x in A & y = s . x ) by A15, FUNCT_1:def_6;
then y in rng s by FUNCT_1:3;
hence y in X by FUNCT_2:def_3; ::_thesis: verum
end;
dom s = X by FUNCT_2:def_1;
then card b = k + 1 by A14, A15, Th4;
hence b in the Lines of (G_ (k,X)) by A9, A16; ::_thesis: verum
end;
then reconsider L = L as Function of the Lines of (G_ (k,X)), the Lines of (G_ (k,X)) by A10, FUNCT_2:2;
take IncProjMap(# P,L #) ; ::_thesis: ( ( for A being POINT of (G_ (k,X)) holds IncProjMap(# P,L #) . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds IncProjMap(# P,L #) . L = s .: L ) )
thus ( ( for A being POINT of (G_ (k,X)) holds IncProjMap(# P,L #) . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds IncProjMap(# P,L #) . L = s .: L ) ) by A1, A10; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict IncProjMap over G_ (k,X), G_ (k,X) st ( for A being POINT of (G_ (k,X)) holds b1 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds b1 . L = s .: L ) & ( for A being POINT of (G_ (k,X)) holds b2 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds b2 . L = s .: L ) holds
b1 = b2
proof
let f1, f2 be strict IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( ( for A being POINT of (G_ (k,X)) holds f1 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds f1 . L = s .: L ) & ( for A being POINT of (G_ (k,X)) holds f2 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds f2 . L = s .: L ) implies f1 = f2 )
assume that
A17: for A being POINT of (G_ (k,X)) holds f1 . A = s .: A and
A18: for L being LINE of (G_ (k,X)) holds f1 . L = s .: L and
A19: for A being POINT of (G_ (k,X)) holds f2 . A = s .: A and
A20: for L being LINE of (G_ (k,X)) holds f2 . L = s .: L ; ::_thesis: f1 = f2
A21: for L being LINE of (G_ (k,X)) holds f1 . L = f2 . L
proof
let L be LINE of (G_ (k,X)); ::_thesis: f1 . L = f2 . L
f1 . L = s .: L by A18;
hence f1 . L = f2 . L by A20; ::_thesis: verum
end;
for A being POINT of (G_ (k,X)) holds f1 . A = f2 . A
proof
let A be POINT of (G_ (k,X)); ::_thesis: f1 . A = f2 . A
f1 . A = s .: A by A17;
hence f1 . A = f2 . A by A19; ::_thesis: verum
end;
hence f1 = f2 by A21, Th16; ::_thesis: verum
end;
end;
:: deftheorem Def14 defines incprojmap COMBGRAS:def_14_:_
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for s being Permutation of X
for b4 being strict IncProjMap over G_ (k,X), G_ (k,X) holds
( b4 = incprojmap (k,s) iff ( ( for A being POINT of (G_ (k,X)) holds b4 . A = s .: A ) & ( for L being LINE of (G_ (k,X)) holds b4 . L = s .: L ) ) );
theorem Th24: :: COMBGRAS:24
for k being Element of NAT
for X being non empty set st k = 1 & k + 1 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
proof
deffunc H1( set ) -> set = {$1};
let k be Element of NAT ; ::_thesis: for X being non empty set st k = 1 & k + 1 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
let X be non empty set ; ::_thesis: ( k = 1 & k + 1 c= card X implies for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume A1: ( k = 1 & k + 1 c= card X ) ; ::_thesis: for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
A2: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = 1 } by A1, Def1;
consider c being Function such that
A3: dom c = X and
A4: for x being set st x in X holds
c . x = H1(x) from FUNCT_1:sch_3();
A5: rng c c= the Points of (G_ (k,X))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng c or y in the Points of (G_ (k,X)) )
assume y in rng c ; ::_thesis: y in the Points of (G_ (k,X))
then consider x being set such that
A6: ( x in dom c & y = c . x ) by FUNCT_1:def_3;
A7: card {x} = 1 by CARD_1:30;
( {x} c= X & y = {x} ) by A3, A4, A6, ZFMISC_1:31;
hence y in the Points of (G_ (k,X)) by A2, A7; ::_thesis: verum
end;
let F be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( F is automorphism implies ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume A8: F is automorphism ; ::_thesis: ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
A9: the point-map of F is bijective by A8, Def9;
reconsider c = c as Function of X, the Points of (G_ (k,X)) by A3, A5, FUNCT_2:2;
deffunc H2( Element of X) -> set = union (F . (c . $1));
consider f being Function such that
A10: dom f = X and
A11: for x being Element of X holds f . x = H2(x) from FUNCT_1:sch_4();
rng f c= X
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng f or b in X )
assume b in rng f ; ::_thesis: b in X
then consider a being set such that
A12: a in X and
A13: b = f . a by A10, FUNCT_1:def_3;
reconsider a = a as Element of X by A12;
A14: b = union (F . (c . a)) by A11, A13;
consider A being POINT of (G_ (k,X)) such that
A15: A = F . (c . a) ;
A in the Points of (G_ (k,X)) ;
then A16: ex A1 being Subset of X st
( A1 = A & card A1 = 1 ) by A2;
then consider x being set such that
A17: A = {x} by CARD_2:42;
x in X by A16, A17, ZFMISC_1:31;
hence b in X by A14, A15, A17, ZFMISC_1:25; ::_thesis: verum
end;
then reconsider f = f as Function of X,X by A10, FUNCT_2:2;
A18: F is incidence_preserving by A8, Def9;
A19: dom the point-map of F = the Points of (G_ (k,X)) by FUNCT_2:52;
A20: f is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom f or not x2 in dom f or not f . x1 = f . x2 or x1 = x2 )
assume that
A21: ( x1 in dom f & x2 in dom f ) and
A22: f . x1 = f . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1, x2 = x2 as Element of X by A21;
consider A1 being POINT of (G_ (k,X)) such that
A23: A1 = F . (c . x1) ;
A1 in the Points of (G_ (k,X)) ;
then ex A11 being Subset of X st
( A11 = A1 & card A11 = 1 ) by A2;
then consider y1 being set such that
A24: A1 = {y1} by CARD_2:42;
A25: ( c . x1 = {x1} & c . x2 = {x2} ) by A4;
consider A2 being POINT of (G_ (k,X)) such that
A26: A2 = F . (c . x2) ;
A2 in the Points of (G_ (k,X)) ;
then ex A12 being Subset of X st
( A12 = A2 & card A12 = 1 ) by A2;
then consider y2 being set such that
A27: A2 = {y2} by CARD_2:42;
f . x2 = union (F . (c . x2)) by A11;
then A28: f . x2 = y2 by A26, A27, ZFMISC_1:25;
f . x1 = union (F . (c . x1)) by A11;
then f . x1 = y1 by A23, A24, ZFMISC_1:25;
then c . x1 = c . x2 by A9, A19, A22, A23, A26, A24, A27, A28, FUNCT_1:def_4;
hence x1 = x2 by A25, ZFMISC_1:3; ::_thesis: verum
end;
A29: rng the point-map of F = the Points of (G_ (k,X)) by A9, FUNCT_2:def_3;
for y being set st y in X holds
ex x being set st
( x in X & y = f . x )
proof
let y be set ; ::_thesis: ( y in X implies ex x being set st
( x in X & y = f . x ) )
assume y in X ; ::_thesis: ex x being set st
( x in X & y = f . x )
then A30: {y} c= X by ZFMISC_1:31;
card {y} = 1 by CARD_1:30;
then {y} in rng the point-map of F by A2, A29, A30;
then consider a being set such that
A31: a in dom the point-map of F and
A32: {y} = the point-map of F . a by FUNCT_1:def_3;
a in the Points of (G_ (k,X)) by A31;
then A33: ex A1 being Subset of X st
( A1 = a & card A1 = 1 ) by A2;
then consider x being set such that
A34: a = {x} by CARD_2:42;
reconsider x = x as Element of X by A33, A34, ZFMISC_1:31;
{y} = F . (c . x) by A4, A32, A34;
then y = union (F . (c . x)) by ZFMISC_1:25;
then y = f . x by A11;
hence ex x being set st
( x in X & y = f . x ) ; ::_thesis: verum
end;
then rng f = X by FUNCT_2:10;
then f is onto by FUNCT_2:def_3;
then reconsider f = f as Permutation of X by A20;
A35: dom the line-map of F = the Lines of (G_ (k,X)) by FUNCT_2:52;
A36: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = 1 + 1 } by A1, Def1;
A37: for x being set st x in dom the line-map of F holds
the line-map of F . x = the line-map of (incprojmap (k,f)) . x
proof
let x be set ; ::_thesis: ( x in dom the line-map of F implies the line-map of F . x = the line-map of (incprojmap (k,f)) . x )
assume A38: x in dom the line-map of F ; ::_thesis: the line-map of F . x = the line-map of (incprojmap (k,f)) . x
then consider A being LINE of (G_ (k,X)) such that
A39: x = A ;
consider A11 being Subset of X such that
A40: x = A11 and
A41: card A11 = 2 by A36, A35, A38;
consider x1, x2 being set such that
A42: x1 <> x2 and
A43: x = {x1,x2} by A40, A41, CARD_2:60;
reconsider x1 = x1, x2 = x2 as Element of X by A40, A43, ZFMISC_1:32;
c . x1 = {x1} by A4;
then consider A1 being POINT of (G_ (k,X)) such that
A44: A1 = {x1} ;
c . x2 = {x2} by A4;
then consider A2 being POINT of (G_ (k,X)) such that
A45: A2 = {x2} ;
A1 <> A2 by A42, A44, A45, ZFMISC_1:18;
then A46: F . A1 <> F . A2 by A9, A19, FUNCT_1:def_4;
F . A2 in the Points of (G_ (k,X)) ;
then A47: ex B2 being Subset of X st
( B2 = F . A2 & card B2 = 1 ) by A2;
then consider y2 being set such that
A48: F . A2 = {y2} by CARD_2:42;
A1 c= A by A39, A43, A44, ZFMISC_1:36;
then A1 on A by A1, Th10;
then F . A1 on F . A by A18, Def8;
then A49: F . A1 c= F . A by A1, Th10;
A50: ( (incprojmap (k,f)) . A = f .: A & f .: (A1 \/ A2) = (f .: A1) \/ (f .: A2) ) by A1, Def14, RELAT_1:120;
A51: A1 \/ A2 = A by A39, A43, A44, A45, ENUMSET1:1;
F . A1 in the Points of (G_ (k,X)) ;
then A52: ex B1 being Subset of X st
( B1 = F . A1 & card B1 = 1 ) by A2;
then A53: ex y1 being set st F . A1 = {y1} by CARD_2:42;
A2 c= A by A39, A43, A45, ZFMISC_1:36;
then A2 on A by A1, Th10;
then F . A2 on F . A by A18, Def8;
then A54: F . A2 c= F . A by A1, Th10;
F . (c . x2) = F . A2 by A4, A45;
then A55: f . x2 = union (F . A2) by A11;
Im (f,x2) = {(f . x2)} by A10, FUNCT_1:59;
then A56: f .: A2 = F . A2 by A45, A55, A48, ZFMISC_1:25;
A57: F . A1 is finite by A52;
not y2 in F . A1
proof
assume y2 in F . A1 ; ::_thesis: contradiction
then F . A2 c= F . A1 by A48, ZFMISC_1:31;
hence contradiction by A46, A52, A47, A57, CARD_FIN:1; ::_thesis: verum
end;
then A58: card ((F . A1) \/ (F . A2)) = 1 + 1 by A52, A53, A48, CARD_2:41;
F . (c . x1) = F . A1 by A4, A44;
then A59: f . x1 = union (F . A1) by A11;
Im (f,x1) = {(f . x1)} by A10, FUNCT_1:59;
then A60: f .: A1 = F . A1 by A44, A59, A53, ZFMISC_1:25;
F . A in the Lines of (G_ (k,X)) ;
then A61: ex B3 being Subset of X st
( B3 = F . A & card B3 = 2 ) by A36;
then F . A is finite ;
hence the line-map of F . x = the line-map of (incprojmap (k,f)) . x by A39, A50, A51, A49, A54, A61, A58, A60, A56, CARD_FIN:1, XBOOLE_1:8; ::_thesis: verum
end;
A62: for x being set st x in dom the point-map of F holds
the point-map of F . x = the point-map of (incprojmap (k,f)) . x
proof
let x be set ; ::_thesis: ( x in dom the point-map of F implies the point-map of F . x = the point-map of (incprojmap (k,f)) . x )
assume A63: x in dom the point-map of F ; ::_thesis: the point-map of F . x = the point-map of (incprojmap (k,f)) . x
then consider A being POINT of (G_ (k,X)) such that
A64: x = A ;
A65: ex A1 being Subset of X st
( x = A1 & card A1 = 1 ) by A2, A19, A63;
then consider x1 being set such that
A66: x = {x1} by CARD_2:42;
reconsider x1 = x1 as Element of X by A65, A66, ZFMISC_1:31;
F . (c . x1) = F . A by A4, A64, A66;
then A67: f . x1 = union (F . A) by A11;
F . A in the Points of (G_ (k,X)) ;
then consider B being Subset of X such that
A68: B = F . A and
A69: card B = 1 by A2;
A70: ex x2 being set st B = {x2} by A69, CARD_2:42;
( (incprojmap (k,f)) . A = f .: A & Im (f,x1) = {(f . x1)} ) by A1, A10, Def14, FUNCT_1:59;
hence the point-map of F . x = the point-map of (incprojmap (k,f)) . x by A64, A66, A67, A68, A70, ZFMISC_1:25; ::_thesis: verum
end;
dom the point-map of (incprojmap (k,f)) = the Points of (G_ (k,X)) by FUNCT_2:52;
then A71: the point-map of F = the point-map of (incprojmap (k,f)) by A19, A62, FUNCT_1:def_11;
dom the line-map of (incprojmap (k,f)) = the Lines of (G_ (k,X)) by FUNCT_2:52;
then IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,f) by A35, A71, A37, FUNCT_1:def_11;
hence ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ; ::_thesis: verum
end;
theorem Th25: :: COMBGRAS:25
for k being Element of NAT
for X being non empty set st 1 < k & card X = k + 1 holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 1 < k & card X = k + 1 holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
let X be non empty set ; ::_thesis: ( 1 < k & card X = k + 1 implies for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume that
A1: 1 < k and
A2: k + 1 = card X ; ::_thesis: for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
deffunc H1( set ) -> Element of bool X = X \ {$1};
consider c being Function such that
A3: dom c = X and
A4: for x being set st x in X holds
c . x = H1(x) from FUNCT_1:sch_3();
A5: X is finite by A2;
A6: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, A2, Def1;
A7: rng c c= the Points of (G_ (k,X))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng c or y in the Points of (G_ (k,X)) )
assume y in rng c ; ::_thesis: y in the Points of (G_ (k,X))
then consider x being set such that
A8: x in dom c and
A9: y = c . x by FUNCT_1:def_3;
A10: card {x} = 1 by CARD_1:30;
{x} c= X by A3, A8, ZFMISC_1:31;
then A11: card (X \ {x}) = (k + 1) - 1 by A2, A5, A10, CARD_2:44;
y = X \ {x} by A3, A4, A8, A9;
hence y in the Points of (G_ (k,X)) by A6, A11; ::_thesis: verum
end;
let F be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( F is automorphism implies ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume F is automorphism ; ::_thesis: ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
then A12: the point-map of F is bijective by Def9;
reconsider c = c as Function of X, the Points of (G_ (k,X)) by A3, A7, FUNCT_2:2;
deffunc H2( Element of X) -> set = union (X \ (F . (c . $1)));
consider f being Function such that
A13: dom f = X and
A14: for x being Element of X holds f . x = H2(x) from FUNCT_1:sch_4();
rng f c= X
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng f or b in X )
assume b in rng f ; ::_thesis: b in X
then consider a being set such that
A15: a in X and
A16: b = f . a by A13, FUNCT_1:def_3;
reconsider a = a as Element of X by A15;
A17: b = union (X \ (F . (c . a))) by A14, A16;
consider A being POINT of (G_ (k,X)) such that
A18: A = F . (c . a) ;
A in the Points of (G_ (k,X)) ;
then ex A1 being Subset of X st
( A1 = A & card A1 = k ) by A6;
then card (X \ A) = (k + 1) - k by A2, A5, CARD_2:44;
then consider x being set such that
A19: X \ A = {x} by CARD_2:42;
x in X by A19, ZFMISC_1:31;
hence b in X by A17, A18, A19, ZFMISC_1:25; ::_thesis: verum
end;
then reconsider f = f as Function of X,X by A13, FUNCT_2:2;
A20: dom the point-map of F = the Points of (G_ (k,X)) by FUNCT_2:52;
A21: f is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom f or not x2 in dom f or not f . x1 = f . x2 or x1 = x2 )
assume that
A22: ( x1 in dom f & x2 in dom f ) and
A23: f . x1 = f . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1, x2 = x2 as Element of X by A22;
consider A1 being POINT of (G_ (k,X)) such that
A24: A1 = F . (c . x1) ;
consider A2 being POINT of (G_ (k,X)) such that
A25: A2 = F . (c . x2) ;
A2 in the Points of (G_ (k,X)) ;
then A26: ex A12 being Subset of X st
( A12 = A2 & card A12 = k ) by A6;
then card (X \ A2) = (k + 1) - k by A2, A5, CARD_2:44;
then consider y2 being set such that
A27: X \ A2 = {y2} by CARD_2:42;
A1 in the Points of (G_ (k,X)) ;
then A28: ex A11 being Subset of X st
( A11 = A1 & card A11 = k ) by A6;
then card (X \ A1) = (k + 1) - k by A2, A5, CARD_2:44;
then consider y1 being set such that
A29: X \ A1 = {y1} by CARD_2:42;
f . x2 = union (X \ (F . (c . x2))) by A14;
then A30: f . x2 = y2 by A25, A27, ZFMISC_1:25;
f . x1 = union (X \ (F . (c . x1))) by A14;
then f . x1 = y1 by A24, A29, ZFMISC_1:25;
then A1 = A2 by A23, A28, A26, A29, A27, A30, Th5;
then A31: c . x1 = c . x2 by A12, A20, A24, A25, FUNCT_1:def_4;
( c . x1 = X \ {x1} & c . x2 = X \ {x2} ) by A4;
then {x1} = {x2} by A31, Th5;
hence x1 = x2 by ZFMISC_1:3; ::_thesis: verum
end;
A32: rng the point-map of F = the Points of (G_ (k,X)) by A12, FUNCT_2:def_3;
for y being set st y in X holds
ex x being set st
( x in X & y = f . x )
proof
let y be set ; ::_thesis: ( y in X implies ex x being set st
( x in X & y = f . x ) )
assume y in X ; ::_thesis: ex x being set st
( x in X & y = f . x )
then A33: {y} c= X by ZFMISC_1:31;
card {y} = 1 by CARD_1:30;
then card (X \ {y}) = (k + 1) - 1 by A2, A5, A33, CARD_2:44;
then X \ {y} in rng the point-map of F by A6, A32;
then consider a being set such that
A34: a in dom the point-map of F and
A35: X \ {y} = the point-map of F . a by FUNCT_1:def_3;
a in the Points of (G_ (k,X)) by A34;
then A36: ex A1 being Subset of X st
( A1 = a & card A1 = k ) by A6;
then card (X \ a) = (k + 1) - k by A2, A5, CARD_2:44;
then consider x being set such that
A37: X \ a = {x} by CARD_2:42;
reconsider x = x as Element of X by A37, ZFMISC_1:31;
a /\ X = X \ {x} by A37, XBOOLE_1:48;
then A38: X \ {x} = a by A36, XBOOLE_1:28;
c . x = X \ {x} by A4;
then X /\ {y} = X \ (F . (c . x)) by A35, A38, XBOOLE_1:48;
then {y} = X \ (F . (c . x)) by A33, XBOOLE_1:28;
then y = union (X \ (F . (c . x))) by ZFMISC_1:25;
then y = f . x by A14;
hence ex x being set st
( x in X & y = f . x ) ; ::_thesis: verum
end;
then A39: rng f = X by FUNCT_2:10;
then f is onto by FUNCT_2:def_3;
then reconsider f = f as Permutation of X by A21;
A40: dom the line-map of F = the Lines of (G_ (k,X)) by FUNCT_2:52;
A41: for x being set st x in dom the point-map of F holds
the point-map of F . x = the point-map of (incprojmap (k,f)) . x
proof
let x be set ; ::_thesis: ( x in dom the point-map of F implies the point-map of F . x = the point-map of (incprojmap (k,f)) . x )
assume A42: x in dom the point-map of F ; ::_thesis: the point-map of F . x = the point-map of (incprojmap (k,f)) . x
then consider A being POINT of (G_ (k,X)) such that
A43: x = A ;
F . A in the Points of (G_ (k,X)) ;
then A44: ex B being Subset of X st
( B = F . A & card B = k ) by A6;
then card (X \ (F . A)) = (k + 1) - k by A2, A5, CARD_2:44;
then A45: ex x2 being set st X \ (F . A) = {x2} by CARD_2:42;
( X \ (X \ (F . A)) = (F . A) /\ X & (F . A) /\ X = F . A ) by A44, XBOOLE_1:28, XBOOLE_1:48;
then A46: F . A = X \ {(union (X \ (F . A)))} by A45, ZFMISC_1:25;
A47: f .: X = X by A39, RELSET_1:22;
A48: ex A1 being Subset of X st
( x = A1 & card A1 = k ) by A6, A20, A42;
then A49: ( X \ (X \ A) = A /\ X & A /\ X = A ) by A43, XBOOLE_1:28, XBOOLE_1:48;
card (X \ A) = (k + 1) - k by A2, A5, A43, A48, CARD_2:44;
then consider x1 being set such that
A50: X \ A = {x1} by CARD_2:42;
reconsider x1 = x1 as Element of X by A50, ZFMISC_1:31;
A51: ( c . x1 = X \ {x1} & Im (f,x1) = {(f . x1)} ) by A4, A13, FUNCT_1:59;
(incprojmap (k,f)) . A = f .: A by A1, A2, Def14;
then (incprojmap (k,f)) . A = (f .: X) \ (f .: {x1}) by A50, A49, FUNCT_1:64;
hence the point-map of F . x = the point-map of (incprojmap (k,f)) . x by A14, A43, A50, A46, A49, A51, A47; ::_thesis: verum
end;
dom the point-map of (incprojmap (k,f)) = the Points of (G_ (k,X)) by FUNCT_2:52;
then A52: the point-map of F = the point-map of (incprojmap (k,f)) by A20, A41, FUNCT_1:def_11;
A53: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A2, Def1;
A54: for x being set st x in dom the line-map of F holds
the line-map of F . x = the line-map of (incprojmap (k,f)) . x
proof
let x be set ; ::_thesis: ( x in dom the line-map of F implies the line-map of F . x = the line-map of (incprojmap (k,f)) . x )
assume A55: x in dom the line-map of F ; ::_thesis: the line-map of F . x = the line-map of (incprojmap (k,f)) . x
then consider A being LINE of (G_ (k,X)) such that
A56: x = A ;
F . A in the Lines of (G_ (k,X)) ;
then ex y being Subset of X st
( y = F . A & card y = k + 1 ) by A53;
then A57: F . A = X by A2, A5, CARD_FIN:1;
ex A11 being Subset of X st
( x = A11 & card A11 = k + 1 ) by A53, A40, A55;
then A58: x = X by A2, A5, CARD_FIN:1;
(incprojmap (k,f)) . A = f .: x by A1, A2, A56, Def14;
hence the line-map of F . x = the line-map of (incprojmap (k,f)) . x by A39, A56, A58, A57, RELSET_1:22; ::_thesis: verum
end;
dom the line-map of (incprojmap (k,f)) = the Lines of (G_ (k,X)) by FUNCT_2:52;
then IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,f) by A40, A52, A54, FUNCT_1:def_11;
hence ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ; ::_thesis: verum
end;
theorem Th26: :: COMBGRAS:26
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for T being Subset of the Points of (G_ (k,X))
for S being Subset of X st card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } holds
S = meet T
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for T being Subset of the Points of (G_ (k,X))
for S being Subset of X st card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } holds
S = meet T
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for T being Subset of the Points of (G_ (k,X))
for S being Subset of X st card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } holds
S = meet T )
assume that
A1: 0 < k and
A2: k + 1 c= card X ; ::_thesis: for T being Subset of the Points of (G_ (k,X))
for S being Subset of X st card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } holds
S = meet T
A3: k - 1 is Element of NAT by A1, NAT_1:20;
let T be Subset of the Points of (G_ (k,X)); ::_thesis: for S being Subset of X st card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } holds
S = meet T
let S be Subset of X; ::_thesis: ( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } implies S = meet T )
assume that
A4: card S = k - 1 and
A5: T = { A where A is Subset of X : ( card A = k & S c= A ) } ; ::_thesis: S = meet T
A6: S is finite by A1, A4, NAT_1:20;
A7: T <> {}
proof
X \ S <> {}
proof
assume X \ S = {} ; ::_thesis: contradiction
then X c= S by XBOOLE_1:37;
then card X = k - 1 by A4, XBOOLE_0:def_10;
then 1 + k <= (- 1) + k by A2, A3, NAT_1:39;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then consider x being set such that
A8: x in X \ S by XBOOLE_0:def_1;
{x} c= X by A8, ZFMISC_1:31;
then A9: ( S c= S \/ {x} & S \/ {x} c= X ) by XBOOLE_1:7, XBOOLE_1:8;
not x in S by A8, XBOOLE_0:def_5;
then card (S \/ {x}) = (k - 1) + 1 by A4, A6, CARD_2:41;
then S \/ {x} in T by A5, A9;
hence T <> {} ; ::_thesis: verum
end;
A10: meet T c= S
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in meet T or y in S )
assume A11: y in meet T ; ::_thesis: y in S
y in S
proof
consider a1 being set such that
A12: a1 in T by A7, XBOOLE_0:def_1;
A13: ex A1 being Subset of X st
( a1 = A1 & card A1 = k & S c= A1 ) by A5, A12;
then A14: a1 is finite ;
X \ a1 <> {}
proof
assume X \ a1 = {} ; ::_thesis: contradiction
then X c= a1 by XBOOLE_1:37;
then card X = k by A13, XBOOLE_0:def_10;
then 1 + k <= 0 + k by A2, NAT_1:39;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then consider y2 being set such that
A15: y2 in X \ a1 by XBOOLE_0:def_1;
assume A16: not y in S ; ::_thesis: contradiction
A17: S misses {y}
proof
assume not S misses {y} ; ::_thesis: contradiction
then S /\ {y} <> {} by XBOOLE_0:def_7;
then consider z being set such that
A18: z in S /\ {y} by XBOOLE_0:def_1;
( z in {y} & z in S ) by A18, XBOOLE_0:def_4;
hence contradiction by A16, TARSKI:def_1; ::_thesis: verum
end;
then S c= a1 \ {y} by A13, XBOOLE_1:86;
then A19: S c= (a1 \ {y}) \/ {y2} by XBOOLE_1:10;
A20: y in a1 by A11, A12, SETFAM_1:def_1;
then y2 <> y by A15, XBOOLE_0:def_5;
then A21: not y in {y2} by TARSKI:def_1;
( card {y} = 1 & {y} c= a1 ) by A20, CARD_1:30, ZFMISC_1:31;
then A22: card (a1 \ {y}) = k - 1 by A13, A14, CARD_2:44;
then not y in a1 \ {y} by A4, A16, A13, A14, A17, CARD_FIN:1, XBOOLE_1:86;
then A23: not y in (a1 \ {y}) \/ {y2} by A21, XBOOLE_0:def_3;
A24: {y2} c= X by A15, ZFMISC_1:31;
a1 \ {y} c= X by A13, XBOOLE_1:1;
then A25: (a1 \ {y}) \/ {y2} c= X by A24, XBOOLE_1:8;
not y2 in a1 \ {y} by A15, XBOOLE_0:def_5;
then card ((a1 \ {y}) \/ {y2}) = (k - 1) + 1 by A14, A22, CARD_2:41;
then (a1 \ {y}) \/ {y2} in T by A5, A25, A19;
hence contradiction by A11, A23, SETFAM_1:def_1; ::_thesis: verum
end;
hence y in S ; ::_thesis: verum
end;
for a1 being set st a1 in T holds
S c= a1
proof
let a1 be set ; ::_thesis: ( a1 in T implies S c= a1 )
assume a1 in T ; ::_thesis: S c= a1
then ex A1 being Subset of X st
( a1 = A1 & card A1 = k & S c= A1 ) by A5;
hence S c= a1 ; ::_thesis: verum
end;
then S c= meet T by A7, SETFAM_1:5;
hence S = meet T by A10, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: COMBGRAS:27
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for T being Subset of the Points of (G_ (k,X)) st T is STAR holds
for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for T being Subset of the Points of (G_ (k,X)) st T is STAR holds
for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } )
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for T being Subset of the Points of (G_ (k,X)) st T is STAR holds
for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } ) )
assume A1: ( 0 < k & k + 1 c= card X ) ; ::_thesis: for T being Subset of the Points of (G_ (k,X)) st T is STAR holds
for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } )
let T be Subset of the Points of (G_ (k,X)); ::_thesis: ( T is STAR implies for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } ) )
assume T is STAR ; ::_thesis: for S being Subset of X st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } )
then consider S1 being Subset of X such that
A2: ( card S1 = k - 1 & T = { A where A is Subset of X : ( card A = k & S1 c= A ) } ) by Def4;
let S be Subset of X; ::_thesis: ( S = meet T implies ( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } ) )
assume A3: S = meet T ; ::_thesis: ( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } )
S1 = meet T by A1, A2, Th26;
hence ( card S = k - 1 & T = { A where A is Subset of X : ( card A = k & S c= A ) } ) by A3, A2; ::_thesis: verum
end;
theorem Th28: :: COMBGRAS:28
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for T1, T2 being Subset of the Points of (G_ (k,X)) st T1 is STAR & T2 is STAR & meet T1 = meet T2 holds
T1 = T2
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for T1, T2 being Subset of the Points of (G_ (k,X)) st T1 is STAR & T2 is STAR & meet T1 = meet T2 holds
T1 = T2
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for T1, T2 being Subset of the Points of (G_ (k,X)) st T1 is STAR & T2 is STAR & meet T1 = meet T2 holds
T1 = T2 )
assume A1: ( 0 < k & k + 1 c= card X ) ; ::_thesis: for T1, T2 being Subset of the Points of (G_ (k,X)) st T1 is STAR & T2 is STAR & meet T1 = meet T2 holds
T1 = T2
let T1, T2 be Subset of the Points of (G_ (k,X)); ::_thesis: ( T1 is STAR & T2 is STAR & meet T1 = meet T2 implies T1 = T2 )
assume that
A2: T1 is STAR and
A3: T2 is STAR and
A4: meet T1 = meet T2 ; ::_thesis: T1 = T2
consider S2 being Subset of X such that
A5: ( card S2 = k - 1 & T2 = { A where A is Subset of X : ( card A = k & S2 c= A ) } ) by A3, Def4;
A6: S2 = meet T2 by A1, A5, Th26;
consider S1 being Subset of X such that
A7: ( card S1 = k - 1 & T1 = { A where A is Subset of X : ( card A = k & S1 c= A ) } ) by A2, Def4;
S1 = meet T1 by A1, A7, Th26;
hence T1 = T2 by A4, A7, A5, A6; ::_thesis: verum
end;
theorem Th29: :: COMBGRAS:29
for k being Element of NAT
for X being non empty set st k + 1 c= card X holds
for A being finite Subset of X st card A = k - 1 holds
^^ (A,X,k) is STAR
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st k + 1 c= card X holds
for A being finite Subset of X st card A = k - 1 holds
^^ (A,X,k) is STAR
let X be non empty set ; ::_thesis: ( k + 1 c= card X implies for A being finite Subset of X st card A = k - 1 holds
^^ (A,X,k) is STAR )
assume A1: k + 1 c= card X ; ::_thesis: for A being finite Subset of X st card A = k - 1 holds
^^ (A,X,k) is STAR
let A be finite Subset of X; ::_thesis: ( card A = k - 1 implies ^^ (A,X,k) is STAR )
assume A2: card A = k - 1 ; ::_thesis: ^^ (A,X,k) is STAR
^^ (A,X,k) = ^^ (A,X) by A1, A2, Def13;
hence ^^ (A,X,k) is STAR by A2, Def4; ::_thesis: verum
end;
theorem Th30: :: COMBGRAS:30
for k being Element of NAT
for X being non empty set st k + 1 c= card X holds
for A being finite Subset of X st card A = k - 1 holds
meet (^^ (A,X,k)) = A
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st k + 1 c= card X holds
for A being finite Subset of X st card A = k - 1 holds
meet (^^ (A,X,k)) = A
let X be non empty set ; ::_thesis: ( k + 1 c= card X implies for A being finite Subset of X st card A = k - 1 holds
meet (^^ (A,X,k)) = A )
assume A1: k + 1 c= card X ; ::_thesis: for A being finite Subset of X st card A = k - 1 holds
meet (^^ (A,X,k)) = A
let A be finite Subset of X; ::_thesis: ( card A = k - 1 implies meet (^^ (A,X,k)) = A )
assume A2: card A = k - 1 ; ::_thesis: meet (^^ (A,X,k)) = A
^^ (A,X,k) = ^^ (A,X) by A1, A2, Def13;
hence meet (^^ (A,X,k)) = A by A1, A2, Th26; ::_thesis: verum
end;
theorem Th31: :: COMBGRAS:31
for k being Element of NAT
for X being non empty set st 0 < k & k + 3 c= card X holds
for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 3 c= card X holds
for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) )
let X be non empty set ; ::_thesis: ( 0 < k & k + 3 c= card X implies for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) ) )
assume that
A1: 0 < k and
A2: k + 3 c= card X ; ::_thesis: for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) )
let F be IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X); ::_thesis: ( F is automorphism implies ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) ) )
assume A3: F is automorphism ; ::_thesis: ex H being IncProjMap over G_ (k,X), G_ (k,X) st
( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) )
0 + 2 < k + (1 + 1) by A1, XREAL_1:6;
then 0 + 2 < (k + 1) + 1 ;
then A4: 2 <= k + 1 by NAT_1:13;
defpred S1[ set , set ] means ex B being finite set st
( B = $1 & $2 = meet (F .: (^^ (B,X,(k + 1)))) );
(k + 1) + 0 <= (k + 1) + 2 by XREAL_1:6;
then k + 1 c= k + 3 by NAT_1:39;
then A5: k + 1 c= card X by A2, XBOOLE_1:1;
then A6: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, Def1;
A7: for e being set st e in the Points of (G_ (k,X)) holds
ex u being set st S1[e,u]
proof
let e be set ; ::_thesis: ( e in the Points of (G_ (k,X)) implies ex u being set st S1[e,u] )
assume e in the Points of (G_ (k,X)) ; ::_thesis: ex u being set st S1[e,u]
then ex B being Subset of X st
( B = e & card B = k ) by A6;
then reconsider B = e as finite Subset of X ;
take meet (F .: (^^ (B,X,(k + 1)))) ; ::_thesis: S1[e, meet (F .: (^^ (B,X,(k + 1))))]
thus S1[e, meet (F .: (^^ (B,X,(k + 1))))] ; ::_thesis: verum
end;
consider Hp being Function such that
A8: dom Hp = the Points of (G_ (k,X)) and
A9: for e being set st e in the Points of (G_ (k,X)) holds
S1[e,Hp . e] from CLASSES1:sch_1(A7);
A10: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, A5, Def1;
(k + 1) + 1 <= (k + 1) + 2 by XREAL_1:6;
then k + 2 c= k + 3 by NAT_1:39;
then A11: (k + 1) + 1 c= card X by A2, XBOOLE_1:1;
then A12: the Points of (G_ ((k + 1),X)) = { A where A is Subset of X : card A = k + 1 } by Def1;
then reconsider Hl = the point-map of F as Function of the Lines of (G_ (k,X)), the Lines of (G_ (k,X)) by A10;
A13: (k + 1) + 2 c= card X by A2;
rng Hp c= the Points of (G_ (k,X))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng Hp or y in the Points of (G_ (k,X)) )
assume y in rng Hp ; ::_thesis: y in the Points of (G_ (k,X))
then consider x being set such that
A14: x in dom Hp and
A15: y = Hp . x by FUNCT_1:def_3;
consider B being finite set such that
A16: B = x and
A17: y = meet (F .: (^^ (B,X,(k + 1)))) by A8, A9, A14, A15;
A18: ex x1 being Subset of X st
( x = x1 & card x1 = k ) by A6, A8, A14;
then card B = (k + 1) - 1 by A16;
then ^^ (B,X,(k + 1)) is STAR by A11, A16, A18, Th29;
then F .: (^^ (B,X,(k + 1))) is STAR by A3, A4, A13, Th23;
then consider S being Subset of X such that
A19: card S = (k + 1) - 1 and
A20: F .: (^^ (B,X,(k + 1))) = { C where C is Subset of X : ( card C = k + 1 & S c= C ) } by Def4;
S = meet (F .: (^^ (B,X,(k + 1)))) by A11, A19, A20, Th26;
hence y in the Points of (G_ (k,X)) by A6, A17, A19; ::_thesis: verum
end;
then reconsider Hp = Hp as Function of the Points of (G_ (k,X)), the Points of (G_ (k,X)) by A8, FUNCT_2:2;
A21: the point-map of F is bijective by A3, Def9;
A22: Hp is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom Hp or not x2 in dom Hp or not Hp . x1 = Hp . x2 or x1 = x2 )
assume that
A23: x1 in dom Hp and
A24: x2 in dom Hp and
A25: Hp . x1 = Hp . x2 ; ::_thesis: x1 = x2
consider X2 being finite set such that
A26: X2 = x2 and
A27: Hp . x2 = meet (F .: (^^ (X2,X,(k + 1)))) by A9, A24;
A28: ex x12 being Subset of X st
( x2 = x12 & card x12 = k ) by A6, A8, A24;
then A29: card X2 = (k + 1) - 1 by A26;
then A30: meet (^^ (X2,X,(k + 1))) = X2 by A11, A26, A28, Th30;
^^ (X2,X,(k + 1)) is STAR by A11, A26, A28, A29, Th29;
then A31: F .: (^^ (X2,X,(k + 1))) is STAR by A3, A4, A13, Th23;
consider X1 being finite set such that
A32: X1 = x1 and
A33: Hp . x1 = meet (F .: (^^ (X1,X,(k + 1)))) by A9, A23;
A34: ex x11 being Subset of X st
( x1 = x11 & card x11 = k ) by A6, A8, A23;
then card X1 = (k + 1) - 1 by A32;
then ^^ (X1,X,(k + 1)) is STAR by A11, A32, A34, Th29;
then A35: F .: (^^ (X1,X,(k + 1))) is STAR by A3, A4, A13, Th23;
meet (^^ (X1,X,(k + 1))) = X1 by A11, A32, A34, A29, Th30;
hence x1 = x2 by A11, A21, A25, A32, A33, A26, A27, A35, A31, A30, Th6, Th28; ::_thesis: verum
end;
take H = IncProjMap(# Hp,Hl #); ::_thesis: ( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) )
A36: dom the point-map of F = the Points of (G_ ((k + 1),X)) by FUNCT_2:52;
A37: H is incidence_preserving
proof
let A1 be POINT of (G_ (k,X)); :: according to COMBGRAS:def_8 ::_thesis: for L1 being LINE of (G_ (k,X)) holds
( A1 on L1 iff H . A1 on H . L1 )
let L1 be LINE of (G_ (k,X)); ::_thesis: ( A1 on L1 iff H . A1 on H . L1 )
A38: S1[A1,Hp . A1] by A9;
L1 in the Lines of (G_ (k,X)) ;
then A39: ex l1 being Subset of X st
( l1 = L1 & card l1 = k + 1 ) by A10;
A1 in the Points of (G_ (k,X)) ;
then consider a1 being Subset of X such that
A40: a1 = A1 and
A41: card a1 = k by A6;
consider L11 being POINT of (G_ ((k + 1),X)) such that
A42: L11 = L1 by A10, A12;
reconsider a1 = a1 as finite Subset of X by A41;
A43: card a1 = (k + 1) - 1 by A41;
A44: ( H . A1 on H . L1 implies A1 on L1 )
proof
( F " (F .: (^^ (a1,X,(k + 1)))) c= ^^ (a1,X,(k + 1)) & ^^ (a1,X,(k + 1)) c= F " (F .: (^^ (a1,X,(k + 1)))) ) by A21, A36, FUNCT_1:76, FUNCT_1:82;
then A45: F " (F .: (^^ (a1,X,(k + 1)))) = ^^ (a1,X,(k + 1)) by XBOOLE_0:def_10;
H . L1 in the Lines of (G_ (k,X)) ;
then A46: ex hl1 being Subset of X st
( hl1 = H . L1 & card hl1 = k + 1 ) by A10;
^^ (a1,X,(k + 1)) is STAR by A11, A43, Th29;
then F .: (^^ (a1,X,(k + 1))) is STAR by A3, A4, A13, Th23;
then consider S being Subset of X such that
A47: card S = (k + 1) - 1 and
A48: F .: (^^ (a1,X,(k + 1))) = { A where A is Subset of X : ( card A = k + 1 & S c= A ) } by Def4;
H . A1 in the Points of (G_ (k,X)) ;
then consider ha1 being Subset of X such that
A49: ha1 = H . A1 and
A50: card ha1 = k by A6;
reconsider ha1 = ha1, S = S as finite Subset of X by A50, A47;
A51: ^^ (ha1,X,(k + 1)) = ^^ (ha1,X) by A11, A50, A47, Def13;
assume H . A1 on H . L1 ; ::_thesis: A1 on L1
then H . A1 c= H . L1 by A1, A5, Th10;
then F . L11 in ^^ (ha1,X,(k + 1)) by A42, A49, A50, A46, A51;
then L1 in F " (^^ (ha1,X,(k + 1))) by A36, A42, FUNCT_1:def_7;
then A52: meet (F " (^^ (ha1,X,(k + 1)))) c= L1 by SETFAM_1:3;
^^ (S,X,(k + 1)) = ^^ (S,X) by A11, A47, Def13;
then A53: S = meet (F .: (^^ (a1,X,(k + 1)))) by A11, A47, A48, Th30;
meet (^^ (a1,X,(k + 1))) = a1 by A11, A41, A47, Th30;
hence A1 on L1 by A1, A5, A40, A38, A49, A50, A48, A51, A53, A52, A45, Th10; ::_thesis: verum
end;
A54: ^^ (a1,X,(k + 1)) = ^^ (a1,X) by A11, A43, Def13;
( A1 on L1 implies H . A1 on H . L1 )
proof
assume A1 on L1 ; ::_thesis: H . A1 on H . L1
then A1 c= L1 by A1, A5, Th10;
then L1 in ^^ (a1,X,(k + 1)) by A40, A41, A39, A54;
then F . L11 in F .: (^^ (a1,X,(k + 1))) by A36, A42, FUNCT_1:def_6;
then meet (F .: (^^ (a1,X,(k + 1)))) c= F . L11 by SETFAM_1:3;
hence H . A1 on H . L1 by A1, A5, A40, A42, A38, Th10; ::_thesis: verum
end;
hence ( A1 on L1 iff H . A1 on H . L1 ) by A44; ::_thesis: verum
end;
A55: rng the point-map of F = the Points of (G_ ((k + 1),X)) by A21, FUNCT_2:def_3;
for y being set st y in the Points of (G_ (k,X)) holds
ex x being set st
( x in the Points of (G_ (k,X)) & y = Hp . x )
proof
let y be set ; ::_thesis: ( y in the Points of (G_ (k,X)) implies ex x being set st
( x in the Points of (G_ (k,X)) & y = Hp . x ) )
assume y in the Points of (G_ (k,X)) ; ::_thesis: ex x being set st
( x in the Points of (G_ (k,X)) & y = Hp . x )
then A56: ex Y1 being Subset of X st
( y = Y1 & card Y1 = k ) by A6;
then reconsider y = y as finite Subset of X ;
A57: card y = (k + 1) - 1 by A56;
then ^^ (y,X,(k + 1)) is STAR by A11, Th29;
then F " (^^ (y,X,(k + 1))) is STAR by A3, A4, A13, Th23;
then consider S being Subset of X such that
A58: card S = (k + 1) - 1 and
A59: F " (^^ (y,X,(k + 1))) = { A where A is Subset of X : ( card A = k + 1 & S c= A ) } by Def4;
A60: S in the Points of (G_ (k,X)) by A6, A58;
reconsider S = S as finite Subset of X by A58;
A61: S1[S,Hp . S] by A9, A60;
^^ (S,X,(k + 1)) = ^^ (S,X) by A11, A58, Def13;
then Hp . S = meet (^^ (y,X,(k + 1))) by A55, A58, A59, A61, FUNCT_1:77;
then y = Hp . S by A11, A57, Th30;
hence ex x being set st
( x in the Points of (G_ (k,X)) & y = Hp . x ) by A60; ::_thesis: verum
end;
then rng Hp = the Points of (G_ (k,X)) by FUNCT_2:10;
then A62: Hp is onto by FUNCT_2:def_3;
A63: for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
Hp . A = meet (F .: (^^ (B,X,(k + 1))))
proof
let A be POINT of (G_ (k,X)); ::_thesis: for B being finite set st B = A holds
Hp . A = meet (F .: (^^ (B,X,(k + 1))))
A64: S1[A,Hp . A] by A9;
let B be finite set ; ::_thesis: ( B = A implies Hp . A = meet (F .: (^^ (B,X,(k + 1)))) )
assume A = B ; ::_thesis: Hp . A = meet (F .: (^^ (B,X,(k + 1))))
hence Hp . A = meet (F .: (^^ (B,X,(k + 1)))) by A64; ::_thesis: verum
end;
the line-map of H is bijective by A3, A10, A12, Def9;
hence ( H is automorphism & the line-map of H = the point-map of F & ( for A being POINT of (G_ (k,X))
for B being finite set st B = A holds
H . A = meet (F .: (^^ (B,X,(k + 1)))) ) ) by A63, A22, A62, A37, Def9; ::_thesis: verum
end;
theorem Th32: :: COMBGRAS:32
for k being Element of NAT
for X being non empty set st 0 < k & k + 3 c= card X holds
for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 3 c= card X holds
for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
let X be non empty set ; ::_thesis: ( 0 < k & k + 3 c= card X implies for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f) )
assume that
A1: 0 < k and
A2: k + 3 c= card X ; ::_thesis: for F being IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X) st F is automorphism holds
for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
k + 1 <= k + 3 by XREAL_1:7;
then k + 1 c= k + 3 by NAT_1:39;
then A3: k + 1 c= card X by A2, XBOOLE_1:1;
then A4: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, Def1;
k + 2 <= k + 3 by XREAL_1:7;
then k + 2 c= k + 3 by NAT_1:39;
then A5: (k + 1) + 1 c= card X by A2, XBOOLE_1:1;
then A6: the Points of (G_ ((k + 1),X)) = { A where A is Subset of X : card A = k + 1 } by Def1;
k + 0 <= k + 1 by XREAL_1:7;
then A7: k c= k + 1 by NAT_1:39;
k + 1 <= k + 2 by XREAL_1:7;
then A8: k + 1 c= k + 2 by NAT_1:39;
let F be IncProjMap over G_ ((k + 1),X), G_ ((k + 1),X); ::_thesis: ( F is automorphism implies for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f) )
assume A9: F is automorphism ; ::_thesis: for H being IncProjMap over G_ (k,X), G_ (k,X) st the line-map of H = the point-map of F holds
for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
A10: F is incidence_preserving by A9, Def9;
let H be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( the line-map of H = the point-map of F implies for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f) )
assume A11: the line-map of H = the point-map of F ; ::_thesis: for f being Permutation of X st IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) holds
IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
A12: dom the point-map of F = the Points of (G_ ((k + 1),X)) by FUNCT_2:52;
let f be Permutation of X; ::_thesis: ( IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) implies IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f) )
assume A13: IncProjMap(# the point-map of H, the line-map of H #) = incprojmap (k,f) ; ::_thesis: IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f)
A14: for x being set st x in dom the point-map of F holds
the point-map of F . x = the point-map of (incprojmap ((k + 1),f)) . x
proof
let x be set ; ::_thesis: ( x in dom the point-map of F implies the point-map of F . x = the point-map of (incprojmap ((k + 1),f)) . x )
assume x in dom the point-map of F ; ::_thesis: the point-map of F . x = the point-map of (incprojmap ((k + 1),f)) . x
then consider A being POINT of (G_ ((k + 1),X)) such that
A15: x = A ;
consider A1 being LINE of (G_ (k,X)) such that
A16: x = A1 by A4, A6, A15;
(incprojmap (k,f)) . A1 = f .: A1 by A1, A3, Def14;
then F . A = (incprojmap ((k + 1),f)) . A by A11, A13, A5, A15, A16, Def14;
hence the point-map of F . x = the point-map of (incprojmap ((k + 1),f)) . x by A15; ::_thesis: verum
end;
A17: the Lines of (G_ ((k + 1),X)) = { L where L is Subset of X : card L = (k + 1) + 1 } by A5, Def1;
A18: the point-map of F is bijective by A9, Def9;
A19: for x being set st x in dom the line-map of F holds
the line-map of F . x = the line-map of (incprojmap ((k + 1),f)) . x
proof
let x be set ; ::_thesis: ( x in dom the line-map of F implies the line-map of F . x = the line-map of (incprojmap ((k + 1),f)) . x )
assume x in dom the line-map of F ; ::_thesis: the line-map of F . x = the line-map of (incprojmap ((k + 1),f)) . x
then consider A being LINE of (G_ ((k + 1),X)) such that
A20: x = A ;
x in the Lines of (G_ ((k + 1),X)) by A20;
then A21: ex A11 being Subset of X st
( x = A11 & card A11 = (k + 1) + 1 ) by A17;
then consider B1 being set such that
A22: B1 c= x and
A23: card B1 = k + 1 by A8, CARD_FIL:36;
A24: x is finite by A21;
then A25: card (x \ B1) = (k + 2) - (k + 1) by A21, A22, A23, CARD_2:44;
B1 c= X by A21, A22, XBOOLE_1:1;
then B1 in the Points of (G_ ((k + 1),X)) by A6, A23;
then consider b1 being POINT of (G_ ((k + 1),X)) such that
A26: b1 = B1 ;
consider C1 being set such that
A27: C1 c= B1 and
A28: card C1 = k by A7, A23, CARD_FIL:36;
B1 is finite by A23;
then A29: card (C1 \/ (x \ B1)) = k + 1 by A27, A28, A24, A25, CARD_2:40, XBOOLE_1:85;
C1 c= x by A22, A27, XBOOLE_1:1;
then A30: C1 \/ (x \ B1) c= x by XBOOLE_1:8;
then C1 \/ (x \ B1) c= X by A21, XBOOLE_1:1;
then C1 \/ (x \ B1) in the Points of (G_ ((k + 1),X)) by A6, A29;
then consider b2 being POINT of (G_ ((k + 1),X)) such that
A31: b2 = C1 \/ (x \ B1) ;
b2 on A by A5, A20, A30, A31, Th10;
then F . b2 on F . A by A10, Def8;
then A32: F . b2 c= F . A by A5, Th10;
B1 \/ (C1 \/ (x \ B1)) c= x by A22, A30, XBOOLE_1:8;
then A33: card (b1 \/ b2) c= k + 2 by A21, A26, A31, CARD_1:11;
B1 misses x \ B1 by XBOOLE_1:79;
then card ((x \ B1) /\ B1) = 0 by CARD_1:27, XBOOLE_0:def_7;
then A34: b1 <> b2 by A25, A26, A31, XBOOLE_1:11, XBOOLE_1:28;
then (k + 1) + 1 c= card (b1 \/ b2) by A23, A29, A26, A31, Th1;
then card (b1 \/ b2) = k + 2 by A33, XBOOLE_0:def_10;
then A35: b1 \/ b2 = x by A21, A22, A24, A30, A26, A31, CARD_FIN:1, XBOOLE_1:8;
F . b2 in the Points of (G_ ((k + 1),X)) ;
then A36: ex B12 being Subset of X st
( F . b2 = B12 & card B12 = k + 1 ) by A6;
F . b1 in the Points of (G_ ((k + 1),X)) ;
then A37: ex B11 being Subset of X st
( F . b1 = B11 & card B11 = k + 1 ) by A6;
F . A in the Lines of (G_ ((k + 1),X)) ;
then A38: ex L1 being Subset of X st
( F . A = L1 & card L1 = (k + 1) + 1 ) by A17;
then A39: F . A is finite ;
F . b1 <> F . b2 by A18, A12, A34, FUNCT_1:def_4;
then A40: (k + 1) + 1 c= card ((F . b1) \/ (F . b2)) by A37, A36, Th1;
b1 on A by A5, A20, A22, A26, Th10;
then F . b1 on F . A by A10, Def8;
then A41: F . b1 c= F . A by A5, Th10;
then (F . b1) \/ (F . b2) c= F . A by A32, XBOOLE_1:8;
then card ((F . b1) \/ (F . b2)) c= k + 2 by A38, CARD_1:11;
then card ((F . b1) \/ (F . b2)) = k + 2 by A40, XBOOLE_0:def_10;
then A42: (F . b1) \/ (F . b2) = F . A by A41, A32, A38, A39, CARD_FIN:1, XBOOLE_1:8;
A43: (incprojmap ((k + 1),f)) . A = f .: x by A5, A20, Def14;
A44: ( (f .: b1) \/ (f .: b2) = f .: (b1 \/ b2) & F . b2 = (incprojmap ((k + 1),f)) . b2 ) by A12, A14, RELAT_1:120;
( F . b1 = (incprojmap ((k + 1),f)) . b1 & (incprojmap ((k + 1),f)) . b1 = f .: b1 ) by A5, A12, A14, Def14;
hence the line-map of F . x = the line-map of (incprojmap ((k + 1),f)) . x by A5, A20, A35, A42, A43, A44, Def14; ::_thesis: verum
end;
A45: ( dom the line-map of F = the Lines of (G_ ((k + 1),X)) & dom the line-map of (incprojmap ((k + 1),f)) = the Lines of (G_ ((k + 1),X)) ) by FUNCT_2:52;
dom the point-map of (incprojmap ((k + 1),f)) = the Points of (G_ ((k + 1),X)) by FUNCT_2:52;
then the point-map of F = the point-map of (incprojmap ((k + 1),f)) by A12, A14, FUNCT_1:def_11;
hence IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((k + 1),f) by A45, A19, FUNCT_1:def_11; ::_thesis: verum
end;
theorem Th33: :: COMBGRAS:33
for k being Element of NAT
for X being non empty set st 2 <= k & k + 2 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 2 <= k & k + 2 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
let X be non empty set ; ::_thesis: ( 2 <= k & k + 2 c= card X implies for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume that
A1: 2 <= k and
A2: k + 2 c= card X ; ::_thesis: for F being IncProjMap over G_ (k,X), G_ (k,X) st F is automorphism holds
ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
defpred S1[ Element of NAT ] means ( 1 <= $1 & $1 <= k implies for F being IncProjMap over G_ ($1,X), G_ ($1,X) st F is automorphism holds
ex f being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ($1,f) );
A3: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A4: S1[i] ; ::_thesis: S1[i + 1]
( 1 <= i + 1 & i + 1 <= k implies for F being IncProjMap over G_ ((i + 1),X), G_ ((i + 1),X) st F is automorphism holds
ex f being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((i + 1),f) )
proof
assume that
1 <= i + 1 and
A5: i + 1 <= k ; ::_thesis: for F being IncProjMap over G_ ((i + 1),X), G_ ((i + 1),X) st F is automorphism holds
ex f being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap ((i + 1),f)
let F2 be IncProjMap over G_ ((i + 1),X), G_ ((i + 1),X); ::_thesis: ( F2 is automorphism implies ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) )
assume A6: F2 is automorphism ; ::_thesis: ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f)
(i + 1) + 2 <= k + 2 by A5, XREAL_1:7;
then A7: i + 3 c= k + 2 by NAT_1:39;
then A8: i + 3 c= card X by A2, XBOOLE_1:1;
A9: ( i = 0 implies ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) )
proof
i + 2 <= i + 3 by XREAL_1:7;
then i + 2 c= i + 3 by NAT_1:39;
then A10: (i + 1) + 1 c= card X by A8, XBOOLE_1:1;
assume i = 0 ; ::_thesis: ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f)
hence ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) by A6, A10, Th24; ::_thesis: verum
end;
( 0 < i implies ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) )
proof
assume A11: 0 < i ; ::_thesis: ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f)
then consider F1 being IncProjMap over G_ (i,X), G_ (i,X) such that
A12: F1 is automorphism and
A13: the line-map of F1 = the point-map of F2 and
for A being POINT of (G_ (i,X))
for B being finite set st B = A holds
F1 . A = meet (F2 .: (^^ (B,X,(i + 1)))) by A2, A6, A7, Th31, XBOOLE_1:1;
0 + 1 < i + 1 by A11, XREAL_1:8;
then consider f being Permutation of X such that
A14: IncProjMap(# the point-map of F1, the line-map of F1 #) = incprojmap (i,f) by A4, A5, A12, NAT_1:13;
IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) by A2, A6, A7, A11, A13, A14, Th32, XBOOLE_1:1;
hence ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) ; ::_thesis: verum
end;
hence ex f being Permutation of X st IncProjMap(# the point-map of F2, the line-map of F2 #) = incprojmap ((i + 1),f) by A9; ::_thesis: verum
end;
hence S1[i + 1] ; ::_thesis: verum
end;
A15: S1[ 0 ] ;
for i being Element of NAT holds S1[i] from NAT_1:sch_1(A15, A3);
then A16: S1[k] ;
let F be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( F is automorphism implies ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
assume F is automorphism ; ::_thesis: ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
hence ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) by A1, A16, XXREAL_0:2; ::_thesis: verum
end;
theorem Th34: :: COMBGRAS:34
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for s being Permutation of X holds incprojmap (k,s) is automorphism
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for s being Permutation of X holds incprojmap (k,s) is automorphism
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for s being Permutation of X holds incprojmap (k,s) is automorphism )
assume A1: ( 0 < k & k + 1 c= card X ) ; ::_thesis: for s being Permutation of X holds incprojmap (k,s) is automorphism
let s be Permutation of X; ::_thesis: incprojmap (k,s) is automorphism
A2: the Points of (G_ (k,X)) = { A where A is Subset of X : card A = k } by A1, Def1;
A3: the point-map of (incprojmap (k,s)) is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom the point-map of (incprojmap (k,s)) or not x2 in dom the point-map of (incprojmap (k,s)) or not the point-map of (incprojmap (k,s)) . x1 = the point-map of (incprojmap (k,s)) . x2 or x1 = x2 )
assume that
A4: x1 in dom the point-map of (incprojmap (k,s)) and
A5: x2 in dom the point-map of (incprojmap (k,s)) and
A6: the point-map of (incprojmap (k,s)) . x1 = the point-map of (incprojmap (k,s)) . x2 ; ::_thesis: x1 = x2
consider X1 being POINT of (G_ (k,X)) such that
A7: X1 = x1 by A4;
x1 in the Points of (G_ (k,X)) by A4;
then A8: ex X11 being Subset of X st
( X11 = x1 & card X11 = k ) by A2;
consider X2 being POINT of (G_ (k,X)) such that
A9: X2 = x2 by A5;
x2 in the Points of (G_ (k,X)) by A5;
then A10: ex X12 being Subset of X st
( X12 = x2 & card X12 = k ) by A2;
A11: (incprojmap (k,s)) . X2 = s .: x2 by A1, A9, Def14;
(incprojmap (k,s)) . X1 = s .: x1 by A1, A7, Def14;
hence x1 = x2 by A6, A7, A9, A8, A10, A11, Th6; ::_thesis: verum
end;
for y being set st y in the Points of (G_ (k,X)) holds
ex x being set st
( x in the Points of (G_ (k,X)) & y = the point-map of (incprojmap (k,s)) . x )
proof
let y be set ; ::_thesis: ( y in the Points of (G_ (k,X)) implies ex x being set st
( x in the Points of (G_ (k,X)) & y = the point-map of (incprojmap (k,s)) . x ) )
assume y in the Points of (G_ (k,X)) ; ::_thesis: ex x being set st
( x in the Points of (G_ (k,X)) & y = the point-map of (incprojmap (k,s)) . x )
then A12: ex B being Subset of X st
( B = y & card B = k ) by A2;
A13: s " y c= dom s by RELAT_1:132;
then A14: s " y c= X by FUNCT_2:52;
rng s = X by FUNCT_2:def_3;
then A15: s .: (s " y) = y by A12, FUNCT_1:77;
then card (s " y) = k by A12, A13, Th4;
then s " y in the Points of (G_ (k,X)) by A2, A14;
then consider A being POINT of (G_ (k,X)) such that
A16: A = s " y ;
y = (incprojmap (k,s)) . A by A1, A15, A16, Def14;
hence ex x being set st
( x in the Points of (G_ (k,X)) & y = the point-map of (incprojmap (k,s)) . x ) ; ::_thesis: verum
end;
then rng the point-map of (incprojmap (k,s)) = the Points of (G_ (k,X)) by FUNCT_2:10;
then A17: the point-map of (incprojmap (k,s)) is onto by FUNCT_2:def_3;
A18: the Lines of (G_ (k,X)) = { L where L is Subset of X : card L = k + 1 } by A1, Def1;
for y being set st y in the Lines of (G_ (k,X)) holds
ex x being set st
( x in the Lines of (G_ (k,X)) & y = the line-map of (incprojmap (k,s)) . x )
proof
let y be set ; ::_thesis: ( y in the Lines of (G_ (k,X)) implies ex x being set st
( x in the Lines of (G_ (k,X)) & y = the line-map of (incprojmap (k,s)) . x ) )
assume y in the Lines of (G_ (k,X)) ; ::_thesis: ex x being set st
( x in the Lines of (G_ (k,X)) & y = the line-map of (incprojmap (k,s)) . x )
then A19: ex B being Subset of X st
( B = y & card B = k + 1 ) by A18;
A20: s " y c= dom s by RELAT_1:132;
then A21: s " y c= X by FUNCT_2:52;
rng s = X by FUNCT_2:def_3;
then A22: s .: (s " y) = y by A19, FUNCT_1:77;
then card (s " y) = k + 1 by A19, A20, Th4;
then s " y in the Lines of (G_ (k,X)) by A18, A21;
then consider A being LINE of (G_ (k,X)) such that
A23: A = s " y ;
y = (incprojmap (k,s)) . A by A1, A22, A23, Def14;
hence ex x being set st
( x in the Lines of (G_ (k,X)) & y = the line-map of (incprojmap (k,s)) . x ) ; ::_thesis: verum
end;
then rng the line-map of (incprojmap (k,s)) = the Lines of (G_ (k,X)) by FUNCT_2:10;
then A24: the line-map of (incprojmap (k,s)) is onto by FUNCT_2:def_3;
A25: dom s = X by FUNCT_2:52;
A26: incprojmap (k,s) is incidence_preserving
proof
let A1 be POINT of (G_ (k,X)); :: according to COMBGRAS:def_8 ::_thesis: for L1 being LINE of (G_ (k,X)) holds
( A1 on L1 iff (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 )
let L1 be LINE of (G_ (k,X)); ::_thesis: ( A1 on L1 iff (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 )
A27: ( s .: A1 = (incprojmap (k,s)) . A1 & s .: L1 = (incprojmap (k,s)) . L1 ) by A1, Def14;
A1 in the Points of (G_ (k,X)) ;
then A28: ex a1 being Subset of X st
( A1 = a1 & card a1 = k ) by A2;
A29: ( (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 implies A1 on L1 )
proof
assume (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 ; ::_thesis: A1 on L1
then s .: A1 c= s .: L1 by A1, A27, Th10;
then A1 c= L1 by A25, A28, FUNCT_1:87;
hence A1 on L1 by A1, Th10; ::_thesis: verum
end;
( A1 on L1 implies (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 )
proof
assume A1 on L1 ; ::_thesis: (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1
then A1 c= L1 by A1, Th10;
then s .: A1 c= s .: L1 by RELAT_1:123;
hence (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 by A1, A27, Th10; ::_thesis: verum
end;
hence ( A1 on L1 iff (incprojmap (k,s)) . A1 on (incprojmap (k,s)) . L1 ) by A29; ::_thesis: verum
end;
the line-map of (incprojmap (k,s)) is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom the line-map of (incprojmap (k,s)) or not x2 in dom the line-map of (incprojmap (k,s)) or not the line-map of (incprojmap (k,s)) . x1 = the line-map of (incprojmap (k,s)) . x2 or x1 = x2 )
assume that
A30: x1 in dom the line-map of (incprojmap (k,s)) and
A31: x2 in dom the line-map of (incprojmap (k,s)) and
A32: the line-map of (incprojmap (k,s)) . x1 = the line-map of (incprojmap (k,s)) . x2 ; ::_thesis: x1 = x2
consider X1 being LINE of (G_ (k,X)) such that
A33: X1 = x1 by A30;
x1 in the Lines of (G_ (k,X)) by A30;
then A34: ex X11 being Subset of X st
( X11 = x1 & card X11 = k + 1 ) by A18;
consider X2 being LINE of (G_ (k,X)) such that
A35: X2 = x2 by A31;
x2 in the Lines of (G_ (k,X)) by A31;
then A36: ex X12 being Subset of X st
( X12 = x2 & card X12 = k + 1 ) by A18;
A37: (incprojmap (k,s)) . X2 = s .: x2 by A1, A35, Def14;
(incprojmap (k,s)) . X1 = s .: x1 by A1, A33, Def14;
hence x1 = x2 by A32, A33, A35, A34, A36, A37, Th6; ::_thesis: verum
end;
hence incprojmap (k,s) is automorphism by A24, A3, A17, A26, Def9; ::_thesis: verum
end;
theorem :: COMBGRAS:35
for k being Element of NAT
for X being non empty set st 0 < k & k + 1 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) holds
( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
proof
let k be Element of NAT ; ::_thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for F being IncProjMap over G_ (k,X), G_ (k,X) holds
( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
let X be non empty set ; ::_thesis: ( 0 < k & k + 1 c= card X implies for F being IncProjMap over G_ (k,X), G_ (k,X) holds
( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ) )
assume that
A1: 0 < k and
A2: k + 1 c= card X ; ::_thesis: for F being IncProjMap over G_ (k,X), G_ (k,X) holds
( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
let F be IncProjMap over G_ (k,X), G_ (k,X); ::_thesis: ( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
A3: ( F is automorphism implies ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) )
proof
A4: ( card k = k & succ 1 = 1 + 1 ) by CARD_1:def_2;
A5: card (k + 1) = k + 1 by CARD_1:def_2;
k + 1 in succ (card X) by A2, ORDINAL1:22;
then A6: ( k + 1 = card X or k + 1 in card X ) by ORDINAL1:8;
A7: card 1 = 1 by CARD_1:def_2;
( 0 + 1 < k + 1 & succ k = k + 1 ) by A1, NAT_1:38, XREAL_1:8;
then 1 in succ k by A7, A5, NAT_1:41;
then ( 1 = k or 1 in k ) by ORDINAL1:8;
then A8: ( 1 = k or ( 1 < k & 2 c= k ) ) by A7, A4, NAT_1:41, ORDINAL1:21;
assume A9: F is automorphism ; ::_thesis: ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s)
succ (k + 1) = (k + 1) + 1 by NAT_1:38;
then ( 1 = k or ( 1 < k & card X = k + 1 ) or ( 2 <= k & k + 2 c= card X ) ) by A6, A8, NAT_1:39, ORDINAL1:21;
hence ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) by A2, A9, Th24, Th25, Th33; ::_thesis: verum
end;
( ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) implies F is automorphism )
proof
assume ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ; ::_thesis: F is automorphism
then consider s being Permutation of X such that
A10: IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ;
A11: incprojmap (k,s) is automorphism by A1, A2, Th34;
then A12: incprojmap (k,s) is incidence_preserving by Def9;
A13: F is incidence_preserving
proof
let A be POINT of (G_ (k,X)); :: according to COMBGRAS:def_8 ::_thesis: for L1 being LINE of (G_ (k,X)) holds
( A on L1 iff F . A on F . L1 )
let L be LINE of (G_ (k,X)); ::_thesis: ( A on L iff F . A on F . L )
( F . A = (incprojmap (k,s)) . A & F . L = (incprojmap (k,s)) . L ) by A10;
hence ( A on L iff F . A on F . L ) by A12, Def8; ::_thesis: verum
end;
( the line-map of F is bijective & the point-map of F is bijective ) by A10, A11, Def9;
hence F is automorphism by A13, Def9; ::_thesis: verum
end;
hence ( F is automorphism iff ex s being Permutation of X st IncProjMap(# the point-map of F, the line-map of F #) = incprojmap (k,s) ) by A3; ::_thesis: verum
end;