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
  f is positive_dilatation & g is negative_dilatation implies f*g is
  negative_dilatation & g*f is negative_dilatation
proof
  assume
A1: f is positive_dilatation & g is negative_dilatation;
  thus x,y // (f*g).y,(f*g).x
  proof
    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 thesis by DIRAF:4;
    end;
    x,y // y9,x9 & y9,x9 // f.y9,f.x9 by A1,Th27;
    hence thesis by A2,A3,DIRAF:3;
  end;
  thus x,y // (g*f).y,(g*f).x
  proof
    set x9=f.x;
    set y9=f.y;
A4: (g*f).x= g.x9 & (g*f).y=g.y9 by FUNCT_2:15;
A5: now
      assume x9=y9;
      then x=y by FUNCT_2:58;
      hence thesis by DIRAF:4;
    end;
    x,y // x9,y9 & x9,y9 // g.y9,g.x9 by A1,Th27;
    hence thesis by A4,A5,DIRAF:3;
  end;
end;
