
theorem Th27: :: Exercise 4.41
  for R, S being RelStr st (the InternalRel of R) c= (the InternalRel of S) &
  R is Dickson & (the carrier of R) = (the carrier of S) holds S is Dickson
proof
  let r, s be RelStr such that
A1: the InternalRel of r c= the InternalRel of s and
A2: r is Dickson and
A3: the carrier of r = the carrier of s;
  let N be Subset of s;
  reconsider N9 = N as Subset of r by A3;
  consider B being set such that
A4: B is_Dickson-basis_of N9,r and
A5: B is finite by A2;
  take B;
  thus B c= N by A4;
  hereby
    let a be Element of s such that
A6: a in N;
    reconsider a9 = a as Element of r by A3;
    consider b being Element of r such that
A7: b in B and
A8: b <= a9 by A4,A6;
    reconsider b9 = b as Element of s by A3;
    take b9;
    [b,a9] in the InternalRel of r by A8;
    hence b9 in B & b9 <= a by A1,A7;
  end;
  thus thesis by A5;
end;
