reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th49:
  AS is not AffinPlane & q in A & q in P & q in C & q<>a & q<>b &
q<>c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line
& P is being_line & C is being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9
  implies b,c // b9,c9
proof
  assume that
A1: AS is not AffinPlane and
A2: q in A and
A3: q in P and
A4: q in C and
A5: q<>a and
A6: q<>b and
A7: q<>c and
A8: a in A & a9 in A and
A9: b in P & b9 in P and
A10: c in C & c9 in C and
A11: A is being_line and
A12: P is being_line and
A13: C is being_line and
A14: A<>P and
A15: A<>C and
A16: a,b // a9,b9 and
A17: a,c // a9,c9;
  now
    assume A,P,C is_coplanar;
    then consider X such that
A18: A c= X and
A19: P c= X and
A20: C c= X and
A21: X is being_plane;
    consider d such that
A22: not d in X by A1,A21,Th48;
    LIN q,a,a9 by A2,A8,A11,AFF_1:21;
    then consider d9 such that
A23: LIN q,d,d9 and
A24: a,d // a9,d9 by A5,Th1;
    set K=Line(q,d);
A25: d in K by AFF_1:15;
    then
A26: not K c= X by A22;
A27: q<>d by A2,A18,A22;
    then
A28: q in K & K is being_line by AFF_1:15,def 3;
    then
A29: d9 in K by A25,A27,A23,AFF_1:25;
    now
      assume
A30:  P<>C;
A31:  not K,P,C is_coplanar
      proof
        assume K,P,C is_coplanar;
        then P,C,K is_coplanar;
        hence contradiction by A12,A13,A19,A20,A21,A26,A30,Th46;
      end;
A32:  K<>A by A18,A22,A25;
      not A,K,P is_coplanar
      proof
        assume A,K,P is_coplanar;
        then A,P,K is_coplanar;
        hence contradiction by A11,A12,A14,A18,A19,A21,A26,Th46;
      end;
      then
A33:  d,b // d9,b9 by A2,A3,A5,A6,A8,A9,A11,A12,A14,A16,A25,A27,A28,A24,A29,A32
,Lm10;
A34:  K<>P & K<>C by A19,A20,A22,A25;
      not A,K,C is_coplanar
      proof
        assume A,K,C is_coplanar;
        then A,C,K is_coplanar;
        hence contradiction by A11,A13,A15,A18,A20,A21,A26,Th46;
      end;
      then
      d,c // d9,c9 by A2,A4,A5,A7,A8,A10,A11,A13,A15,A17,A25,A27,A28,A24,A29
,A32,Lm10;
      hence thesis by A3,A4,A6,A7,A9,A10,A12,A13,A25,A27,A28,A29,A34,A31,A33
,Lm10;
    end;
    hence thesis by A9,A10,A12,AFF_1:51;
  end;
  hence thesis by A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,Lm10;
end;
