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;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;

theorem Th44:
  f is dilatation & g is dilatation implies f*g is dilatation
proof
  assume
A1: f is dilatation & g is dilatation;
  now
    let x,y;
    set x9=g.x;
    set y9=g.y;
A2: (f*g).x= f.x9 & (f*g).y=f.y9 by FUNCT_2:15;
A3: now
      assume x9=y9;
      then x=y by FUNCT_2:58;
      hence x,y '||' (f*g).x,(f*g).y by DIRAF:20;
    end;
    x,y '||' x9,y9 & x9,y9 '||' f.x9,f.y9 by A1,Th34;
    hence x,y '||' (f*g).x,(f*g).y by A2,A3,DIRAF:23;
  end;
  hence thesis by Th34;
end;
