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 Th54:
  not LIN a,b,c & a9<>b9 & a,b // a9,b9 implies ex c9 st a,c // a9
  ,c9 & b,c // b9,c9
proof
  assume that
A1: not LIN a,b,c and
A2: a9<>b9 and
A3: a,b // a9,b9;
  now
    consider X such that
A4: a in X and
A5: b in X and
A6: c in X and
A7: X is being_plane by Th37;
    assume
A8: AS is not AffinPlane;
    now
      set A=Line(a,a9),P=Line(b,b9);
      set AB=Line(a,b),AB9=Line(a9,b9);
A9:   a in AB by AFF_1:15;
      assume
A10:  not a9 in X;
      then
A11:  A is being_line by A4,AFF_1:def 3;
A12:  a<>b by A1,AFF_1:7;
      then
A13:  AB c= X by A4,A5,A7,Th19;
A14:  AB // AB9 by A2,A3,A12,AFF_1:37;
      then consider Y such that
A15:  AB c= Y and
A16:  AB9 c= Y and
A17:  Y is being_plane by Th39;
A18:  b in AB by AFF_1:15;
A19:  a9 in AB9 by AFF_1:15;
      then
A20:  A c= Y by A4,A10,A9,A15,A16,A17,Th19;
A21:  b9 in AB9 by AFF_1:15;
A22:  not b9 in X
      proof
        assume b9 in X;
        then AB9 c= X by A7,A21,A14,A13,Th23;
        hence contradiction by A10,A19;
      end;
      then
A23:  P is being_line by A5,AFF_1:def 3;
A24:  b9 in P by AFF_1:15;
A25:  a in A by AFF_1:15;
A26:  a<>c by A1,AFF_1:7;
A27:  b in P by AFF_1:15;
A28:  a9 in A by AFF_1:15;
A29:  AB is being_line by A12,AFF_1:def 3;
A30:  A<>P
      proof
        assume A=P;
        then A=AB by A12,A9,A18,A29,A11,A25,A27,AFF_1:18;
        hence contradiction by A10,A13,A28;
      end;
A31:  now
        set C=c*A;
        assume
A32:    A // P;
A33:    c in C by A11,Def3;
A34:    A<>C
        proof
          assume A=C;
          then A=Line(a,c) by A26,A11,A25,A33,AFF_1:57;
          then A c= X by A4,A6,A7,A26,Th19;
          hence contradiction by A10,A28;
        end;
A35:    A // C by A11,Def3;
        then consider c9 such that
A36:    c9 in C and
A37:    a,c // a9,c9 by A25,A28,A33,Lm2;
        C is being_line by A11,Th27;
        then b,c // b9,c9 by A3,A8,A11,A23,A25,A28,A27,A24,A30,A32,A33,A35,A36
,A37,A34,Th51;
        hence thesis by A37;
      end;
A38:  a9 in Y by A19,A16;
A39:  now
        given q such that
A40:    q in A and
A41:    q in P;
A42:    q<>a
        proof
          assume q=a;
          then AB=P by A12,A9,A18,A29,A23,A27,A41,AFF_1:18;
          hence contradiction by A13,A22,A24;
        end;
A43:    q<>b
        proof
          assume q=b;
          then AB=A by A12,A9,A18,A29,A11,A25,A40,AFF_1:18;
          hence contradiction by A10,A13,A28;
        end;
        set C=Line(q,c);
A44:    c in C by AFF_1:15;
A45:    A<>C
        proof
          assume A=C;
          then A=Line(a,c) by A26,A11,A25,A44,AFF_1:57;
          then A c= X by A4,A6,A7,A26,Th19;
          hence contradiction by A10,A28;
        end;
        LIN q,a,a9 by A11,A25,A28,A40,AFF_1:21;
        then consider c9 such that
A46:    LIN q,c,c9 and
A47:    a,c // a9,c9 by A42,Th1;
A48:    q<>c by A1,A4,A5,A6,A7,A10,A9,A18,A15,A17,A38,A20,A40,Th25;
        then
A49:    q in C & C is being_line by AFF_1:15,def 3;
        then c9 in C by A48,A44,A46,AFF_1:25;
        then
        b,c // b9,c9 by A3,A8,A11,A23,A25,A28,A27,A24,A30,A40,A41,A42,A43,A48
,A44,A49,A47,A45,Th49;
        hence thesis by A47;
      end;
      P c= Y by A5,A18,A21,A22,A15,A16,A17,Th19;
      hence thesis by A17,A11,A23,A20,A31,A39,Th22;
    end;
    hence thesis by A1,A3,A4,A5,A6,A7,Lm12;
  end;
  hence thesis by A1,Th53;
end;
