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 Th51:
  AS is not AffinPlane & A // P & A // 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: A // P and
A3: A // C and
A4: a in A & a9 in A and
A5: b in P & b9 in P and
A6: c in C & c9 in C and
A7: A is being_line and
A8: P is being_line and
A9: C is being_line and
A10: A<>P and
A11: A<>C and
A12: a,b // a9,b9 and
A13: a,c // a9,c9;
  now
    assume A,P,C is_coplanar;
    then consider X such that
A14: A c= X and
A15: P c= X and
A16: C c= X and
A17: X is being_plane;
    consider d such that
A18: not d in X by A1,A17,Th48;
    set K=d*A;
A19: d in K by A7,Def3;
    then
A20: not K c= X by A18;
A21: A // K by A7,Def3;
    ex d9 st d9 in K & a,d // a9,d9
    proof
A22:  now
        assume
A23:    a<>a9;
        consider d9 such that
A24:    a,a9 // d,d9 and
A25:    a,d // a9,d9 by DIRAF:40;
        d,d9 // a,a9 by A24,AFF_1:4;
        then d,d9 // A by A4,A7,A23,AFF_1:27;
        then d,d9 // K by A21,Th3;
        then d9 in K by A19,Th2;
        hence thesis by A25;
      end;
      now
        assume
A26:    a=a9;
        take d9=d;
        thus d9 in K by A7,Def3;
        thus a,d // a9,d9 by A26,AFF_1:2;
      end;
      hence thesis by A22;
    end;
    then consider d9 such that
A27: d9 in K and
A28: a,d // a9,d9;
A29: K // P & K // C by A2,A3,A21,AFF_1:44;
    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 A8,A9,A15,A16,A17,A20,A30,Th46;
      end;
A32:  K<>A by A14,A18,A19;
      not A,K,P is_coplanar
      proof
        assume A,K,P is_coplanar;
        then A,P,K is_coplanar;
        hence contradiction by A7,A8,A10,A14,A15,A17,A20,Th46;
      end;
      then
A33:  d,b // d9,b9 by A2,A4,A5,A7,A10,A12,A19,A21,A27,A28,A32,Lm11;
A34:  K<>P & K<>C by A15,A16,A18,A19;
      not A,K,C is_coplanar
      proof
        assume A,K,C is_coplanar;
        then A,C,K is_coplanar;
        hence contradiction by A7,A9,A11,A14,A16,A17,A20,Th46;
      end;
      then d,c // d9,c9 by A3,A4,A6,A7,A11,A13,A19,A21,A27,A28,A32,Lm11;
      hence thesis by A5,A6,A7,A19,A29,A27,A34,A31,A33,Lm11,Th27;
    end;
    hence thesis by A5,A6,A8,AFF_1:51;
  end;
  hence thesis by A2,A3,A4,A5,A6,A7,A10,A11,A12,A13,Lm11;
end;
