let OAS be OAffinSpace; :: thesis: for a, b, c being Element of OAS ex x being Element of OAS st
( a,x // b,c & a,b // x,c )
let a, b, c be Element of OAS; :: thesis: ex x being Element of OAS st
( a,x // b,c & a,b // x,c )

A1: now

consider e3 being Element of OAS such that
A2: Mid b,c,e3 and
A3: c <> e3 by DIRAF:13;
A4: b,c // c,e3 by A2, DIRAF:def 3;
assume A5: not LIN a,b,c ; :: thesis: ex x being Element of OAS st
( a,x // b,c & a,b // x,c )
then A6: b <> c by DIRAF:31;
then consider e4 being Element of OAS such that
A7: a,c // c,e4 and
A8: a,b // e3,e4 by A4, ANALOAF:def 5;
A9: Mid a,c,e4 by A7, DIRAF:def 3;
then Mid e4,c,a by DIRAF:9;
then A10: e4,c // c,a by DIRAF:def 3;
A11: e4 <> c

proof

assume A12: e4 = c ; :: thesis: contradiction
e3,c // c,b by A4, DIRAF:2;
then a,b // c,b by A3, A8, A12, DIRAF:3;
then b,a // b,c by DIRAF:2;
then ( Mid b,a,c or Mid b,c,a ) by DIRAF:7;
then ( LIN b,a,c or LIN b,c,a ) by DIRAF:28;
hence contradiction by A5, DIRAF:30; :: thesis: verum

end;

A13: not LIN e4,c,e3

proof

LIN b,c,e3 by A2, DIRAF:28;
then A14: LIN c,e3,b by DIRAF:30;
assume A15: LIN e4,c,e3 ; :: thesis: contradiction
A16: LIN c,e3,c by DIRAF:31;
A17: LIN e4,c,c by DIRAF:31;
LIN e4,c,a by A9, DIRAF:9, DIRAF:28;
then LIN c,e3,a by A11, A15, A17, DIRAF:32;
hence contradiction by A5, A3, A14, A16, DIRAF:32; :: thesis: verum

end;

b,c // c,e3 by A2, DIRAF:def 3;
then A18: c,e3 // b,c by DIRAF:2;
consider e1 being Element of OAS such that
A19: Mid c,a,e1 and
A20: a <> e1 by DIRAF:13;
A21: c,a // a,e1 by A19, DIRAF:def 3;
A22: a <> c by A5, DIRAF:31;
then consider e2 being Element of OAS such that
A23: b,a // a,e2 and
A24: b,c // e1,e2 by A21, ANALOAF:def 5;
c,a // c,e1 by A21, ANALOAF:def 5;
then e4,c // c,e1 by A22, A10, DIRAF:3;
then A25: Mid e4,c,e1 by DIRAF:def 3;
then consider e5 being Element of OAS such that
A26: Mid e4,e3,e5 and
A27: c,e3 // e1,e5 by A11, Th27;
A28: e4 <> e3

proof

assume e4 = e3 ; :: thesis: contradiction
then Mid a,c,e3 by A7, DIRAF:def 3;
then A29: LIN e3,c,a by DIRAF:9, DIRAF:28;
A30: LIN e3,c,c by DIRAF:31;
LIN e3,c,b by A2, DIRAF:9, DIRAF:28;
hence contradiction by A5, A3, A29, A30, DIRAF:32; :: thesis: verum

end;

then A31: e4 <> e5 by A26, DIRAF:8;
A32: e1 <> e4 by A25, A11, DIRAF:8;
A33: not LIN e1,e4,e3

proof

LIN e4,c,e1 by A25, DIRAF:28;
then A34: LIN e4,e1,c by DIRAF:30;
assume LIN e1,e4,e3 ; :: thesis: contradiction
then A35: LIN e4,e1,e3 by DIRAF:30;
LIN e4,e1,e4 by DIRAF:31;
hence contradiction by A32, A13, A35, A34, DIRAF:32; :: thesis: verum

end;

A36: not LIN e1,e5,e4

proof

LIN e4,e3,e5 by A26, DIRAF:28;
then A37: LIN e5,e4,e3 by DIRAF:30;
assume LIN e1,e5,e4 ; :: thesis: contradiction
then A38: LIN e5,e4,e1 by DIRAF:30;
LIN e5,e4,e4 by DIRAF:31;
hence contradiction by A31, A33, A38, A37, DIRAF:32; :: thesis: verum

end;

then A39: e1 <> e5 by DIRAF:31;
Mid e1,c,e4 by A25, DIRAF:9;
then consider e6 being Element of OAS such that
A40: Mid e1,e6,e5 and
A41: e5,e4 // e6,c by A36, Th28;
A42: c <> e1 by A19, A20, DIRAF:8;
A43: not LIN c,e1,b

proof

LIN c,a,e1 by A19, DIRAF:28;
then A44: LIN c,e1,a by DIRAF:30;
A45: LIN c,e1,c by DIRAF:31;
assume LIN c,e1,b ; :: thesis: contradiction
hence contradiction by A5, A42, A44, A45, DIRAF:32; :: thesis: verum

end;

A46: e5 <> e3

proof

assume e5 = e3 ; :: thesis: contradiction
then e3,c // e3,e1 by A27, DIRAF:2;
then ( Mid e3,c,e1 or Mid e3,e1,c ) by DIRAF:7;
then ( LIN e3,c,e1 or LIN e3,e1,c ) by DIRAF:28;
then A47: LIN e3,c,e1 by DIRAF:30;
A48: LIN e3,c,c by DIRAF:31;
LIN e3,c,b by A2, DIRAF:9, DIRAF:28;
hence contradiction by A3, A43, A47, A48, DIRAF:32; :: thesis: verum

end;

A49: e1 <> e6

proof

Mid e5,e3,e4 by A26, DIRAF:9;
then A50: e5,e3 // e3,e4 by DIRAF:def 3;
then e5,e3 // e5,e4 by ANALOAF:def 5;
then A51: e5,e4 // e3,e4 by A46, A50, ANALOAF:def 5;
assume e1 = e6 ; :: thesis: contradiction
then e3,e4 // e1,c by A31, A41, A51, ANALOAF:def 5;
then A52: a,b // e1,c by A8, A28, DIRAF:3;
Mid e1,a,c by A19, DIRAF:9;
then A53: e1,a // a,c by DIRAF:def 3;
then e1,a // e1,c by ANALOAF:def 5;
then e1,c // a,c by A20, A53, ANALOAF:def 5;
then a,b // a,c by A42, A52, DIRAF:3;
then ( a,b // b,c or a,c // c,b ) by DIRAF:6;
then ( Mid a,b,c or Mid a,c,b ) by DIRAF:def 3;
then ( LIN a,b,c or LIN a,c,b ) by DIRAF:28;
hence contradiction by A5, DIRAF:30; :: thesis: verum

end;

consider x being Element of OAS such that
A54: Mid c,x,e6 and
A55: e1,e6 // a,x by A19, Th26;
e1,e6 // e6,e5 by A40, DIRAF:def 3;
then e1,e6 // e1,e5 by ANALOAF:def 5;
then e1,e5 // a,x by A55, A49, ANALOAF:def 5;
then c,e3 // a,x by A27, A39, DIRAF:3;
then A56: a,x // b,c by A3, A18, ANALOAF:def 5;
A57: e6 <> c

proof

assume e6 = c ; :: thesis: contradiction
then x = c by A54, DIRAF:8;
then c,a // c,b by A56, DIRAF:2;
then ( Mid c,a,b or Mid c,b,a ) by DIRAF:7;
then ( LIN c,a,b or LIN c,b,a ) by DIRAF:28;
hence contradiction by A5, DIRAF:30; :: thesis: verum

end;

A58: a <> e2

proof

assume A59: a = e2 ; :: thesis: contradiction
e1,a // a,c by A21, DIRAF:2;
then b,c // a,c by A20, A24, A59, DIRAF:3;
then c,b // c,a by DIRAF:2;
then ( Mid c,b,a or Mid c,a,b ) by DIRAF:7;
then ( LIN c,b,a or LIN c,a,b ) by DIRAF:28;
hence contradiction by A5, DIRAF:30; :: thesis: verum

end;

A60: e6 <> x

proof

assume e6 = x ; :: thesis: contradiction
then e6,e1 // e6,a by A55, DIRAF:2;
then ( Mid e6,e1,a or Mid e6,a,e1 ) by DIRAF:7;
then ( LIN e6,e1,a or LIN e6,a,e1 ) by DIRAF:28;
then A61: LIN e6,e1,a by DIRAF:30;
LIN e1,e6,e5 by A40, DIRAF:28;
then A62: LIN e6,e1,e5 by DIRAF:30;
b,c // e1,e5 by A3, A4, A27, DIRAF:3;
then e1,e2 // e1,e5 by A6, A24, ANALOAF:def 5;
then ( Mid e1,e2,e5 or Mid e1,e5,e2 ) by DIRAF:7;
then ( LIN e1,e2,e5 or LIN e1,e5,e2 ) by DIRAF:28;
then A63: LIN e1,e5,e2 by DIRAF:30;
A64: LIN e1,e5,e1 by DIRAF:31;
Mid b,a,e2 by A23, DIRAF:def 3;
then LIN b,a,e2 by DIRAF:28;
then A65: LIN a,e2,b by DIRAF:30;
LIN c,a,e1 by A19, DIRAF:28;
then A66: LIN a,e1,c by DIRAF:30;
LIN e6,e1,e1 by DIRAF:31;
then LIN e1,e5,a by A49, A61, A62, DIRAF:32;
then A67: LIN a,e1,e2 by A39, A63, A64, DIRAF:32;
A68: LIN a,e2,a by DIRAF:31;
LIN a,e1,a by DIRAF:31;
then LIN a,e2,c by A20, A67, A66, DIRAF:32;
hence contradiction by A5, A58, A65, A68, DIRAF:32; :: thesis: verum

end;

Mid e6,x,c by A54, DIRAF:9;
then A69: e6,x // x,c by DIRAF:def 3;
then e6,x // e6,c by ANALOAF:def 5;
then A70: e6,c // x,c by A60, A69, ANALOAF:def 5;
Mid e5,e3,e4 by A26, DIRAF:9;
then A71: e5,e3 // e3,e4 by DIRAF:def 3;
then e5,e3 // e5,e4 by ANALOAF:def 5;
then e3,e4 // e5,e4 by A46, A71, ANALOAF:def 5;
then a,b // e5,e4 by A8, A28, DIRAF:3;
then a,b // e6,c by A31, A41, DIRAF:3;
then a,b // x,c by A57, A70, DIRAF:3;
hence ex x being Element of OAS st
( a,x // b,c & a,b // x,c ) by A56; :: thesis: verum

end;

now

A72: now

assume Mid a,c,b ; :: thesis: ( a,a // b,c & a,b // a,c )
then a,c // c,b by DIRAF:def 3;
then a,c // a,b by ANALOAF:def 5;
hence ( a,a // b,c & a,b // a,c ) by DIRAF:2, DIRAF:4; :: thesis: verum

end;

A73: now

assume Mid b,a,c ; :: thesis: ( a,c // b,c & a,b // c,c )
then Mid c,a,b by DIRAF:9;
then c,a // a,b by DIRAF:def 3;
then c,a // c,b by ANALOAF:def 5;
hence ( a,c // b,c & a,b // c,c ) by DIRAF:2, DIRAF:4; :: thesis: verum

end;

A74: ( Mid a,b,c implies ( a,b // b,c & a,b // b,c ) ) by DIRAF:def 3;
assume LIN a,b,c ; :: thesis: ex x being Element of OAS st
( a,x // b,c & a,b // x,c )
hence ex x being Element of OAS st
( a,x // b,c & a,b // x,c ) by A74, A73, A72, DIRAF:29; :: thesis: verum

end;

hence ex x being Element of OAS st
( a,x // b,c & a,b // x,c ) by A1; :: thesis: verum