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 Th26:
  x=a & [X,1]=A implies (a on A iff X is being_line & x in X)
proof
  assume that
A1: x=a and
A2: [X,1]=A;
A3: now
A4: now
      given K such that
A5:   K is being_line and
A6:   [X,1]=[K,1] and
A7:   x in the carrier of AS & x in K or x = LDir(K);
A8:   now
        assume x=LDir(K);
        then x in Dir_of_Lines(AS) by A5,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;
      X=[K,1]`1 by A6
        .= K;
      hence X is being_line & x in X by A5,A7,A8;
    end;
    assume a on A;
    then
A9: [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
    not ex K,X9 st K is being_line & X9 is being_plane & x=LDir(K) & [X,1
    ]=[PDir(X9),2] & K '||' X9 by XTUPLE_0:1;
    hence X is being_line & x in X by A1,A2,A9,A4,Def11;
  end;
  now
    assume that
A10: X is being_line and
A11: x in X;
    [x,[X,1]] in Proj_Inc(AS) by A10,A11,Def11;
    hence a on A by A1,A2,INCSP_1:def 1;
  end;
  hence thesis by A3;
end;
