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 Th30:
  X is being_line & a=LDir(X) & A=[X,1] implies a on A
proof
  assume that
A1: X is being_line and
A2: a=LDir(X) and
A3: A=[X,1];
  X // X by A1,AFF_1:41;
  then X '||' X by A1,AFF_4:40;
  hence thesis by A1,A2,A3,Th28;
end;
