
theorem Th45:
for G being SimpleGraph, L being set
 holds Vertices (G SubgraphInducedBy L) = (Vertices G) /\ L
proof
let G be SimpleGraph, L be set;
 set S= G SubgraphInducedBy L; set uS = union S; set uG = union G;
  union (G /\ bool L) c= union G /\ union bool L by ZFMISC_1:79;
 hence uS c= uG /\ L by ZFMISC_1:81;
 thus uG /\ L c= uS proof
   let a be object;
   assume a in uG /\ L;
    then a in uG & a in L by XBOOLE_0:def 4;
   hence a in uS by Lm8;
 end;
end;
