let G be IncProjStr ; :: thesis: for a, b, c being POINT of G

for P, Q, R being LINE of G holds

( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

let a, b, c be POINT of G; :: thesis: for P, Q, R being LINE of G holds

( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

let P, Q, R be LINE of G; :: thesis: ( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

thus ( {a,b} on P implies {b,a} on P ) ; :: thesis: ( ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

thus ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) :: thesis: ( ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

assume A3: a on P,Q,R ; :: thesis: ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P )

then A4: a on R ;

( a on P & a on Q ) by A3;

hence ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) by A4; :: thesis: verum

for P, Q, R being LINE of G holds

( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

let a, b, c be POINT of G; :: thesis: for P, Q, R being LINE of G holds

( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

let P, Q, R be LINE of G; :: thesis: ( ( {a,b} on P implies {b,a} on P ) & ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

thus ( {a,b} on P implies {b,a} on P ) ; :: thesis: ( ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) & ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

thus ( {a,b,c} on P implies ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) ) :: thesis: ( ( a on P,Q implies a on Q,P ) & ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) ) )

proof

thus
( a on P,Q implies a on Q,P )
; :: thesis: ( a on P,Q,R implies ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) )
assume A1:
{a,b,c} on P
; :: thesis: ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P )

then A2: c on P by INCSP_1:2;

( a on P & b on P ) by A1, INCSP_1:2;

hence ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) by A2, INCSP_1:2; :: thesis: verum

end;then A2: c on P by INCSP_1:2;

( a on P & b on P ) by A1, INCSP_1:2;

hence ( {a,c,b} on P & {b,a,c} on P & {b,c,a} on P & {c,a,b} on P & {c,b,a} on P ) by A2, INCSP_1:2; :: thesis: verum

assume A3: a on P,Q,R ; :: thesis: ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P )

then A4: a on R ;

( a on P & a on Q ) by A3;

hence ( a on P,R,Q & a on Q,P,R & a on Q,R,P & a on R,P,Q & a on R,Q,P ) by A4; :: thesis: verum