
theorem Th6: :: DClique:
for R being RelStr, S being Subset of R holds S is Clique of R iff
for a, b being Element of R st a in S & b in S & a <> b holds a <= b or b <= a
proof
  let R be RelStr, S be Subset of R;
  set RR = the InternalRel of R;
  hereby
    assume S is Clique of R;
    then A1: RR is_connected_in S by Def1;
    let a, b be Element of R;
    assume a in S & b in S & a <> b;
     then [a,b] in RR or [b,a] in RR by A1;
    hence a <= b or b <= a;
  end;
  assume
  A2: for a, b being Element of R st a in S & b in S & a <> b
      holds a <= b or b <= a;
  RR is_connected_in S proof
    let x, y be object;
    assume A3: x in S & y in S & x <> y;
    then reconsider x9 = x, y9 = y as Element of R;
    x9 <= y9 or y9 <= x9 by A3,A2;
    hence [x,y] in RR or [y,x] in RR;
  end;
  hence S is Clique of R by Def1;
end;
