theorem
  f is dilatation & g is dilatation & f.a=g.a & f.b=g.b implies a=b or f =g
proof
  assume that
A1: f is dilatation and
A2: g is dilatation and
A3: f.a=g.a and
A4: f.b=g.b;
A5: (g"*f).b=g".(g.b) by A4,FUNCT_2:15
    .=b by Th2;
A6: g" is dilatation by A2,Th43;
  assume
A7: a<>b;
  (g"*f).a=g".(g.a) by A3,FUNCT_2:15
    .=a by Th2;
  then
A8: g"*f=(id the carrier of OAS) by A1,A7,A5,A6,Th44,Th51;
  now
    let x;
    g".(f.x)=(g"*f).x by FUNCT_2:15;
    then g".(f.x)=x by A8;
    hence g.x=f.x by Th2;
  end;
  hence thesis by FUNCT_2:63;
end;
