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;

theorem Th5:
  not A // K & A '||' X & A '||' Y & K '||' X & K '||' Y & A is
  being_line & K is being_line & X is being_plane & Y is being_plane implies X
  '||' Y
proof
  assume that
A1: not A // K and
A2: A '||' X and
A3: A '||' Y and
A4: K '||' X and
A5: K '||' Y and
A6: A is being_line and
A7: K is being_line and
A8: X is being_plane and
A9: Y is being_plane;
  set y = the Element of Y;
  set x = the Element of X;
A10: Y <> {} by A9,AFF_4:59;
A11: X <> {} by A8,AFF_4:59;
  then reconsider a=x,b=y as Element of AS by A10,TARSKI:def 3;
A12: K // a*K by A7,AFF_4:def 3;
A13: A // a*A by A6,AFF_4:def 3;
A14: not a*A // a*K
  proof
    assume not thesis;
    then a*A // K by A12,AFF_1:44;
    hence contradiction by A1,A13,AFF_1:44;
  end;
  a*K c= a+X by A4,A7,A8,AFF_4:68;
  then
A15: a*K c= X by A8,A11,AFF_4:66;
  K // b*K by A7,AFF_4:def 3;
  then
A16: a*K // b*K by A12,AFF_1:44;
  b*A c= b+Y by A3,A6,A9,AFF_4:68;
  then
A17: b*A c= Y by A9,A10,AFF_4:66;
  A // b*A by A6,AFF_4:def 3;
  then
A18: a*A // b*A by A13,AFF_1:44;
  b*K c= b+Y by A5,A7,A9,AFF_4:68;
  then
A19: b*K c= Y by A9,A10,AFF_4:66;
  a*A c= a+X by A2,A6,A8,AFF_4:68;
  then a*A c= X by A8,A11,AFF_4:66;
  hence thesis by A8,A9,A15,A17,A19,A14,A18,A16,AFF_4:55;
end;
