reserve i, I for set,
  f, g, h for Function,
  s for ManySortedSet of I;
reserve G1, G2, G3 for non empty multMagma,
  x1, x2 for Element of G1,
  y1, y2 for Element of G2,
  z1, z2 for Element of G3;
reserve G1, G2, G3 for Group-like non empty multMagma;

theorem
  1_product <*G1*> = <*1_G1*>
proof
  set s = <*1_G1*>, f = <*G1*>;
  dom s = {1} by FINSEQ_1:2,def 8;
  then reconsider s as ManySortedSet of {1} by PARTFUN1:def 2,RELAT_1:def 18;
  for i being set st i in {1} ex G being Group-like non empty multMagma
  st G = f.i & s.i = 1_G
  proof
    let i be set;
    assume i in {1};
    then
A1: i = 1 by TARSKI:def 1;
    then reconsider G = f.i as Group-like non empty multMagma;
    take G;
    thus thesis by A1;
  end;
  hence thesis by Th5;
end;
