
theorem Th45: :: Wsubp1:
for R being with_finite_stability# RelStr, A being StableSet of R,
    S being Subset of R st card A = stability# R & A c= S
 holds stability# subrelstr S = stability# R
proof
 let R be with_finite_stability# RelStr, A be StableSet of R, S be Subset of R
 such that
A1: card A = stability# R and
A2: A c= S;
   A = A /\ S by A2,XBOOLE_1:28; then
A3: A is StableSet of subrelstr S by Th31;
   consider As being StableSet of subrelstr S such that
A4: card(As) = stability# subrelstr S by Def6;
A5:  card A  <= card As by A3,A4,Def6;
   stability# subrelstr S <= stability# R by Th44;
 hence stability# subrelstr S = stability# R by A4,A1,A5,XXREAL_0:1;
end;
