reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem Th50:
  for A,P,C being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st o in A & o in P & o in C & o<>a & o<>b & o<>c & a 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 & (o=a9 or a=a9) holds b
  ,c // b9,c9
proof
  let A,P,C be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
  such that
A1: o in A and
A2: o in P and
A3: o in C and
A4: o<>a and
A5: o<>b and
A6: o<>c and
A7: a in A and
A8: b in P and
A9: b9 in P and
A10: c in C and
A11: c9 in C and
A12: A is being_line and
A13: P is being_line and
A14: C is being_line and
A15: A<>P and
A16: A<>C and
A17: a,b // a9,b9 and
A18: a,c // a9,c9 and
A19: o=a9 or a=a9;
A20: now
    assume
A21: a=a9;
    then
A22: c =c9 by A1,A3,A4,A6,A7,A10,A11,A12,A14,A16,A18,AFF_4:9;
    b=b9 by A1,A2,A4,A5,A7,A8,A9,A12,A13,A15,A17,A21,AFF_4:9;
    hence thesis by A22,AFF_1:2;
  end;
  now
    assume
A23: o=a9;
    then
A24: o=c9 by A1,A3,A4,A6,A7,A10,A11,A12,A14,A16,A18,AFF_4:8;
    o=b9 by A1,A2,A4,A5,A7,A8,A9,A12,A13,A15,A17,A23,AFF_4:8;
    hence thesis by A24,AFF_1:3;
  end;
  hence thesis by A19,A20;
end;
