
theorem Th3:
  for G1, G2, H1, H2 being non empty multMagma, f being Function of
  G1,H1, g being Function of G2,H2 st f is one-to-one & g is one-to-one holds
  Gr2Iso(f,g) is one-to-one
proof
  let G1, G2, H1, H2 be non empty multMagma;
  let f be Function of G1,H1, g be Function of G2,H2 such that
A1: f is one-to-one and
A2: g is one-to-one;
  let a, b be object;
  set h = Gr2Iso(f,g);
  assume a in dom h;
  then consider a1 being Element of G1, a2 being Element of G2 such that
A3: a = <*a1,a2*> and
A4: h.a = <*f.a1,g.a2*> by Def1;
  assume b in dom h;
  then consider b1 being Element of G1, b2 being Element of G2 such that
A5: b = <*b1,b2*> and
A6: h.b = <*f.b1,g.b2*> by Def1;
  assume
A7: h.a = h.b;
  then dom f = the carrier of G1 & f.a1 = f.b1 by A4,A6,FINSEQ_1:77
,FUNCT_2:def 1;
  then
A8: a1 = b1 by A1;
  dom g = the carrier of G2 & g.a2 = g.b2 by A4,A6,A7,FINSEQ_1:77,FUNCT_2:def 1
;
  hence thesis by A2,A3,A5,A8;
end;
