
theorem Th37:
for  x, y being set holds CompleteSGraph {x,y} = {{},{x},{y},{x,y}}
proof
  let x, y be set;
  thus CompleteSGraph {x,y} c= {{},{x},{y},{x,y}} proof
   let a be object;
   assume a in CompleteSGraph {x,y};
   then consider V being finite Subset of {x,y} such that
  A1: a = V and card V <= 2;
     a = {} or a = {x} or a = {y} or a = {x,y} by A1,ZFMISC_1:36;
   hence thesis by ENUMSET1:def 2;
  end;
A2: {x,y} = Vertices CompleteSGraph {x,y} by Lm1;
A3: x in {x,y} by TARSKI:def 2;
A4: y in {x,y} by TARSKI:def 2;
A5: card {x,y} <= 2 by CARD_2:50;
  thus {{},{x},{y},{x,y}} c= CompleteSGraph {x,y} proof
    let a be object;
    assume a in {{},{x},{y},{x,y}};
    then a = {} or a = {x} or a = {y} or a = {x,y} by ENUMSET1:def 2;
    hence thesis by A2,A3,A4,A5,Th20,Th24;
  end;
end;
