
theorem Th16:
  for S being non empty non void TopStruct for f being
  Collineation of S for X being Subset of S st X is strong holds f.:X is strong
proof
  let S be non empty non void TopStruct;
  let f be Collineation of S;
  let X be Subset of S;
  assume
A1: X is strong;
  thus f.:X is strong
  proof
    let a,b be Point of S;
    assume that
A2: a in f.:X and
A3: b in f.:X;
    thus a,b are_collinear
    proof
      per cases;
      suppose
        a=b;
        hence thesis;
      end;
      suppose
A4:     a<>b;
        consider B being object such that
A5:     B in dom f and
A6:     B in X and
A7:     b = f.B by A3,FUNCT_1:def 6;
        consider A being object such that
A8:     A in dom f and
A9:     A in X and
A10:    a = f.A by A2,FUNCT_1:def 6;
        reconsider A,B as Point of S by A8,A5;
        A,B are_collinear by A1,A9,A6;
        then consider l being Block of S such that
A11:    {A,B} c= l by A4,A10,A7;
        B in l by A11,ZFMISC_1:32;
        then
A12:    b in f.:l by A5,A7,FUNCT_1:def 6;
        A in l by A11,ZFMISC_1:32;
        then a in f.:l by A8,A10,FUNCT_1:def 6;
        then {a,b} c= f.:l by A12,ZFMISC_1:32;
        hence thesis;
      end;
    end;
  end;
end;
