
theorem Th46:
for G being SimpleGraph, x being set
 st x in Vertices G holds G SubgraphInducedBy {x} = { {}, {x} }
proof
  let G be SimpleGraph, x being set such that
A1: x in Vertices G;
   set Gx = G SubgraphInducedBy {x};
  thus Gx c= { {}, {x} } proof
    let a be object;
    assume a in Gx;
    then a in bool {x} by XBOOLE_0:def 4;
    then a = {} or a = {x} by ZFMISC_1:33;
   hence thesis by TARSKI:def 2;
  end;
  thus { {}, {x} } c= Gx proof
    let a be object;
  reconsider aa=a as set by TARSKI:1;
  A2: {} in G & {x} in G by Th20,A1,Th24;
    assume a in { {}, {x} };
    then A3: a = {} or a = {x} by TARSKI:def 2;
    then aa c= {x};
    hence thesis by A2,A3,XBOOLE_0:def 4;
  end;
end;
