reserve

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

theorem Th20:
  for S being IncProjStr for F being IncProjMap over S,S for K
  being Subset of the Points of S holds F is incidence_preserving & K is clique
  implies F.:K is 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 incidence_preserving and
A2: K is clique;
  let B1,B2 be POINT of S;
  assume that
A3: B1 in F.:K and
A4: B2 in F.:K;
A5: F.:K = {B where B is POINT of S:ex A being POINT of S st (A in K & F.A =
  B)} by Th18;
  then consider B11 being POINT of S such that
A6: B1 = B11 and
A7: ex A being POINT of S st A in K & F.A = B11 by A3;
  consider B12 being POINT of S such that
A8: B2 = B12 and
A9: ex A being POINT of S st A in K & F.A = B12 by A5,A4;
  consider A12 being POINT of S such that
A10: A12 in K and
A11: F.A12 = B12 by A9;
  consider A11 being POINT of S such that
A12: A11 in K and
A13: F.A11 = B11 by A7;
  consider L1 being LINE of S such that
A14: {A11,A12} on L1 by A2,A12,A10;
  A12 on L1 by A14,INCSP_1:1;
  then
A15: F.A12 on F.L1 by A1;
  A11 on L1 by A14,INCSP_1:1;
  then F.A11 on F.L1 by A1;
  then {B1,B2} on F.L1 by A6,A8,A13,A11,A15,INCSP_1:1;
  hence thesis;
end;
