reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th22:
  for S being IncProjStr for F being IncProjMap over S,S for K
  being Subset of the Points of S holds F is automorphism & K is maximal_clique
  implies F.:K is maximal_clique & F"K is maximal_clique
proof
  let S be IncProjStr;
  let F be IncProjMap over S,S;
  let K be Subset of the Points of S;
  assume that
A1: F is automorphism and
A2: K is maximal_clique;
A3: F is incidence_preserving by A1;
  the point-map of F is bijective by A1;
  then
A4: the Points of S = rng(the point-map of F) by FUNCT_2:def 3;
A5: the Points of S = dom (the point-map of F) by FUNCT_2:52;
A6: for U being Subset of the Points of S st U is clique & F"K c= U holds U
  = F"K
  proof
    let U be Subset of the Points of S such that
A7: U is clique and
A8: F"K c= U;
    F.:(F"K) c= F.:U by A8,RELAT_1:123;
    then
A9: K c= F.:U by A4,FUNCT_1:77;
A10: U c= F"(F.:U) by A5,FUNCT_1:76;
    F.:U is clique by A3,A7,Th20;
    then U c= F"K by A2,A9,A10;
    hence thesis by A8,XBOOLE_0:def 10;
  end;
A11: the line-map of F is bijective by A1;
A12: for U being Subset of the Points of S st U is clique & F.:K c= U holds
  U = F.:K
  proof
A13: K c= F"(F.:K) by A5,FUNCT_1:76;
    let U be Subset of the Points of S such that
A14: U is clique and
A15: F.:K c= U;
    F"(F.:K) c= F"U by A15,RELAT_1:143;
    then
A16: K c= F"U by A13;
    F"U is clique by A11,A3,A14,Th21;
    then F.:(F"U) c= F.:K by A2,A16;
    then U c= F.:K by A4,FUNCT_1:77;
    hence thesis by A15,XBOOLE_0:def 10;
  end;
A17: K is clique by A2;
  then
A18: F.:K is clique by A3,Th20;
  F"K is clique by A11,A17,A3,Th21;
  hence thesis by A18,A12,A6;
end;
