reserve x,y,z for object, X for set, I for non empty set, i,j for Element of I,
    M0 for multMagma-yielding Function,
    M for non empty multMagma-yielding Function,
    M1, M2, M3 for non empty multMagma,
    G for Group-like multMagma-Family of I,
    H for Group-like associative multMagma-Family of I;
reserve p, q for FinSequence of FreeAtoms(H), g,h for Element of H.i,
  k for Nat;
reserve s,t for Element of FreeProduct(H);

theorem Th46:
  for I being empty set, H being Group-like associative multMagma-Family of I
  holds the carrier of FreeProduct(H) = { 1_FreeProduct(H) }
proof
  let I be empty set, H be Group-like associative multMagma-Family of I;
  now
    let x be object;
    hereby
      assume x in the carrier of FreeProduct(H);
      then A1: x in Class EqCl ReductionRel H;
      reconsider X = x as set by TARSKI:1;
      consider y being object such that
        A2: y in the carrier of FreeAtoms(H)*+^+<0> &
          X = Class(EqCl ReductionRel H,y) by A1, EQREL_1:def 3;
      y in FreeAtoms(H)* by A2, MONOID_0:61;
      then y in {{}} by FUNCT_7:17;
      then y = {} by TARSKI:def 1;
      hence x = 1_FreeProduct(H) by A2, Th45;
    end;
    assume x = 1_FreeProduct(H);
    hence x in the carrier of FreeProduct(H);
  end;
  hence thesis by TARSKI:def 1;
end;
