
theorem Th56:
for G being SimpleGraph, x, y being set
 st {x,y} in G holds G SubgraphInducedBy {x,y} is Clique of G
proof
 let G be SimpleGraph, x, y be set such that
A1: {x,y} in G;
   set S = G SubgraphInducedBy {x,y};
   now
     let a, b be set such that
   A2: a in union S and
   A3: b in union S;
   A4: a in {x,y} & b in {x,y} by A2,A3,Lm7;
       then A5: (a = x or a = y) & (b = x or b = y) by TARSKI:def 2;
     per cases;
     suppose a = b;
       then {a,b} = {a} by ENUMSET1:29;
       hence {a,b} in S by A2,Th24;
     end;
     suppose a <> b;
       hence {a,b} in S by A4,A5,A1,Lm10;
     end;
    end;
   then S = CompleteSGraph Vertices S by Th32;
  hence G SubgraphInducedBy {x,y} is Clique of G;
end;
