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);

theorem Th27:
  x=a & [PDir(X),2]=A implies not a on A
proof
  assume that
A1: x=a and
A2: [PDir(X),2]=A;
A3: now
    given K such that
    K is being_line and
A4: [PDir(X),2]=[K,1] and
    x in the carrier of AS & x in K or x = LDir(K);
    2 = [K,1]`2 by A4
      .= 1;
    hence contradiction;
  end;
A5: now
    given K,X9 such that
A6: K is being_line and
    X9 is being_plane and
A7: x=LDir(K) and
    [PDir(X),2]=[PDir(X9),2] and
    K '||' X9;
    x in Dir_of_Lines(AS) by A6,A7,Th14;
    then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by XBOOLE_0:def 4;
    then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
    hence contradiction by Th16;
  end;
  assume a on A;
  then [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
  hence contradiction by A1,A2,A3,A5,Def11;
end;
