reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;

theorem Th25:
  for f,g being Permutation of the carrier of CS st f is_DIL_of CS
  & g is_DIL_of CS holds f*g is_DIL_of CS
proof
  let f,g be Permutation of the carrier of CS;
A1: now
    let a,b,x,y,z,t be Element of CS;
    assume that
A2: [[x,y],[a,b]] in the CONGR of CS & [[a,b],[z,t]] in the CONGR of CS and
A3: a<>b;
    x,y // a,b & a,b // z,t by A2,ANALOAF:def 2;
    then x,y // z,t by A3,Def5;
    hence [[x,y],[z,t]] in the CONGR of CS by ANALOAF:def 2;
  end;
A4: now
    let x,y,z be Element of CS;
    x,x // y,z by Def5;
    hence [[x,x],[y,z]] in the CONGR of CS by ANALOAF:def 2;
  end;
  assume f is_DIL_of CS & g is_DIL_of CS;
  then f is_FormalIz_of the CONGR of CS & g is_FormalIz_of the CONGR of CS;
  then f*g is_FormalIz_of the CONGR of CS by A1,A4,Th15;
  hence thesis;
end;
