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 Th48:
  for M,N being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st M is being_line & N is being_line & M<>N & o in M & o in N
  & o<>b & o<>b9 & o<>c9 & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9
  // b,a9 & b,c9 // c,b9 & (a=b or b=c or a=c) holds a,c9 // c,a9
proof
  let M,N be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
  such that
A1: M is being_line and
A2: N is being_line and
A3: M<>N and
A4: o in M and
A5: o in N and
A6: o<>b and
A7: o<>b9 and
A8: o<>c9 and
A9: b in M and
A10: c in M and
A11: a9 in N and
A12: b9 in N and
A13: c9 in N and
A14: a,b9 // b,a9 and
A15: b,c9 // c,b9 and
A16: a=b or b=c or a=c;
A17: c <>b9 by A1,A2,A3,A4,A5,A7,A10,A12,AFF_1:18;
  now
    assume
A18: a=c;
    then b,c9 // b,a9 by A14,A15,A17,AFF_1:5;
    then a9=c9 by A1,A2,A3,A4,A5,A6,A8,A9,A11,A13,AFF_4:9;
    hence thesis by A18,AFF_1:2;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A11,A12,A13,A14,A15,A16,AFF_4:9;
end;
