reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th16:
  K // M implies Plane(K,P) = Plane(M,P)
proof
  assume
A1: K // M;
  now
    let x be object;
A2: now
      assume x in Plane(M,P);
      then consider a such that
A3:   x=a and
A4:   ex b st a,b // M & b in P;
      consider b such that
A5:   a,b // M and
A6:   b in P by A4;
      a,b // K by A1,A5,AFF_1:43;
      hence x in Plane(K,P) by A3,A6;
    end;
    now
      assume x in Plane(K,P);
      then consider a such that
A7:   x=a and
A8:   ex b st a,b // K & b in P;
      consider b such that
A9:   a,b // K and
A10:  b in P by A8;
      a,b // M by A1,A9,AFF_1:43;
      hence x in Plane(M,P) by A7,A10;
    end;
    hence x in Plane(K,P) iff x in Plane(M,P) by A2;
  end;
  hence thesis by TARSKI:2;
end;
