let AFS be AffinSpace; :: thesis: for a, b, c, d being Element of AFS holds
( not a,b // c,d or a,c // b,d or ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
let a, b, c, d be Element of AFS; :: thesis: ( not a,b // c,d or a,c // b,d or ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
assume A1: a,b // c,d ; :: thesis: ( a,c // b,d or ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
assume A2: not a,c // b,d ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
A3: now :: thesis: ( a <> b implies ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
consider z being Element of AFS such that
A4: a,b // c,z and
A5: a,c // b,z by DIRAF:40;
assume A6: a <> b ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
A7: now :: thesis: ( b <> z implies ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
c,d // c,z by A1, A6, A4, AFF_1:5;
then LIN c,d,z by AFF_1:def 1;
then LIN d,c,z by AFF_1:6;
then d,c // d,z by AFF_1:def 1;
then z,d // d,c by AFF_1:4;
then consider t being Element of AFS such that
A8: b,d // d,t and
A9: b,z // c,t by A2, A5, DIRAF:40;
assume b <> z ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
then a,c // c,t by A5, A9, AFF_1:5;
then c,a // c,t by AFF_1:4;
then LIN c,a,t by AFF_1:def 1;
then A10: LIN a,c,t by AFF_1:6;
d,b // d,t by A8, AFF_1:4;
then LIN d,b,t by AFF_1:def 1;
then LIN b,d,t by AFF_1:6;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) by A10; :: thesis: verum
end;
now :: thesis: ( b = z implies ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
assume b = z ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
then b,a // b,c by A4, AFF_1:4;
then LIN b,a,c by AFF_1:def 1;
then A11: LIN a,c,b by AFF_1:6;
LIN b,d,b by AFF_1:7;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) by A11; :: thesis: verum
end;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) by A7; :: thesis: verum
end;
now :: thesis: ( a = b implies ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) )
assume a = b ; :: thesis: ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x )
then ( LIN a,c,a & LIN b,d,a ) by AFF_1:7;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) ; :: thesis: verum
end;
hence ex x being Element of AFS st
( LIN a,c,x & LIN b,d,x ) by A3; :: thesis: verum