reserve

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

theorem Th21:
  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 & the line-map
  of F is onto & 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: the line-map of F is onto and
A3: K is clique;
  let A1,A2 be POINT of S;
  assume A1 in F"K & A2 in F"K;
  then F.A1 in K & F.A2 in K by FUNCT_1:def 7;
  then consider L2 being LINE of S such that
A4: {F.A1,F.A2} on L2 by A3;
  the Lines of S = rng(the line-map of F) by A2,FUNCT_2:def 3;
  then consider l1 being object such that
A5: l1 in dom(the line-map of F) and
A6: L2 = (the line-map of F).l1 by FUNCT_1:def 3;
  consider L1 being LINE of S such that
A7: L1 = l1 by A5;
A8: L2 = F.L1 by A6,A7;
  F.A2 on L2 by A4,INCSP_1:1;
  then
A9: A2 on L1 by A1,A8;
  F.A1 on L2 by A4,INCSP_1:1;
  then A1 on L1 by A1,A8;
  then {A1,A2} on L1 by A9,INCSP_1:1;
  hence thesis;
end;
