
theorem :: Hsubp1:
for R being with_finite_clique# RelStr, C being Clique of R,
    S being Subset of R st card C = clique# R & C c= S
 holds clique# subrelstr S = clique# R
proof
 let R be with_finite_clique# RelStr, C be Clique of R, S be Subset of R
 such that
A1: card C = clique# R and
A2: C c= S;
   C = C /\ S by A2,XBOOLE_1:28; then
A3: C is Clique of subrelstr S by Th29;
   consider Cs being Clique of subrelstr S such that
A4: card(Cs) = clique# subrelstr S by Def4;
A5:  card C  <= card Cs by A3,A4,Def4;
   clique# subrelstr S <= clique# R by Th42;
 hence clique# subrelstr S = clique# R by A4,A1,A5,XXREAL_0:1;
end;
