reserve

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

theorem Th16:
  for S1,S2 being IncProjStr for F1,F2 being IncProjMap over S1,S2
st (for A being POINT of S1 holds F1.A = F2.A) & (for L being LINE of S1 holds
  F1.L = F2.L) holds the IncProjMap of F1 = the IncProjMap of F2
proof
  let S1,S2 be IncProjStr;
  let F1,F2 be IncProjMap over S1,S2;
  assume that
A1: for A being POINT of S1 holds F1.A = F2.A and
A2: for L being LINE of S1 holds F1.L = F2.L;
  for a being object holds (a in the Points of S1 implies (the point-map of
  F1).a = (the point-map of F2).a)
  proof
    let a be object;
    assume a in the Points of S1;
    then consider A being POINT of S1 such that
A3: A = a;
    F1.A = F2.A by A1;
    hence thesis by A3;
  end;
  then
A4: the point-map of F1 = the point-map of F2 by FUNCT_2:12;
  for l being object holds (l in the Lines of S1 implies (the line-map of F1)
  .l = (the line-map of F2).l)
  proof
    let l be object;
    assume l in the Lines of S1;
    then consider L being LINE of S1 such that
A5: L = l;
    F1.L = F2.L by A2;
    hence thesis by A5;
  end;
  hence thesis by A4,FUNCT_2:12;
end;
