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 Th41:
  for x,y being Element of the Points of ProjHorizon(AS), X being
  Element of the Lines of IncProjSp_of(AS) st x<>y & [x,X] in the Inc of
  IncProjSp_of(AS) & [y,X] in the Inc of IncProjSp_of(AS) ex Y being Element of
  the Lines of ProjHorizon(AS) st X=[Y,2]
proof
  let x,y be Element of the Points of ProjHorizon(AS), X be Element of the
  Lines of IncProjSp_of(AS);
  reconsider a=x,b=y as POINT of IncProjSp_of(AS) by Th22;
  assume that
A1: x<>y and
A2: [x,X] in the Inc of IncProjSp_of(AS) and
A3: [y,X] in the Inc of IncProjSp_of(AS);
A4: b on X by A3,INCSP_1:def 1;
  consider Y being Element of the Lines of ProjHorizon(AS) such that
A5: x on Y and
A6: y on Y by Th40;
  reconsider A=[Y,2] as LINE of IncProjSp_of(AS) by Th25;
  consider Z being Subset of AS such that
A7: Y=PDir(Z) and
A8: Z is being_plane by Th15;
  consider X2 being Subset of AS such that
A9: y=LDir(X2) and
A10: X2 is being_line by Th14;
  X2 '||' Z by A9,A10,A6,A7,A8,Th36;
  then
A11: b on A by A9,A10,A7,A8,Th29;
  take Y;
  consider X1 being Subset of AS such that
A12: x=LDir(X1) and
A13: X1 is being_line by Th14;
  X1 '||' Z by A12,A13,A5,A7,A8,Th36;
  then
A14: a on A by A12,A13,A7,A8,Th29;
  a on X by A2,INCSP_1:def 1;
  hence thesis by A1,A4,A14,A11,Lm2;
end;
