
theorem Th43:
for G being SimpleGraph, L being set
 holds G SubgraphInducedBy L = G SubgraphInducedBy (L /\ Vertices G)
proof
 let G be SimpleGraph, L be set;
 thus G SubgraphInducedBy L c= G SubgraphInducedBy (L /\ Vertices G) proof
   let x be object;
  reconsider xx=x as set by TARSKI:1;
   assume A1: x in G SubgraphInducedBy L;
       then A2: x in bool L by XBOOLE_0:def 4;
   A3: xx c= Vertices G by A1,ZFMISC_1:74;
   A4: xx c= L /\ Vertices G by A2,A3,XBOOLE_1:19;
   thus x in G SubgraphInducedBy (L /\ Vertices G) by A1,A4,XBOOLE_0:def 4;
 end;
 thus G SubgraphInducedBy (L /\ Vertices G) c= G SubgraphInducedBy L proof
   let x be object;
  reconsider xx=x as set by TARSKI:1;
   assume A5: x in G SubgraphInducedBy (L /\ Vertices G);
       then x in bool (L /\ Vertices G) by XBOOLE_0:def 4; then
   A6: xx c= L by XBOOLE_1:18;
   thus thesis by A5,A6,XBOOLE_0:def 4;
 end;
end;
