reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

theorem
  F is commutative associative & F is having_a_unity & e = the_unity_wrt
  F & f.:B = {e} implies F$$(B,f) = e
proof
  assume that
A1: F is commutative associative and
A2: F is having_a_unity and
A3: e = the_unity_wrt F;
  defpred X[Element of Fin C] means f.:($1) = {e} implies F$$($1,f) = e;
A4: for B9 being Element of Fin C, b being Element of C holds X[B9] & not b
  in B9 implies X[B9 \/ {.b.}]
  proof
    let B9,c such that
A5: f.:(B9) = {e} implies F$$(B9,f) = e and
A6: not c in B9 and
A7: f.:(B9 \/ {c}) = {e};
A8: now
      per cases;
      suppose
        B9 = {};
        then
A9:     B9 = {}.C;
        thus F$$(B9 \/ {.c.},f) = F.(F $$(B9,f),f.c) by A1,A2,A6,Th2
          .= F.(e,f.c) by A1,A2,A3,A9,SETWISEO:31;
      end;
      suppose
A10:    B9 <> {};
        B9 c= C by FINSUB_1:def 5;
        then
A11:    B9 c= dom f by FUNCT_2:def 1;
        f.:B9 c= {e} by A7,RELAT_1:123,XBOOLE_1:7;
        hence F$$(B9 \/ {.c.},f) = F.(e,f.c) by A1,A5,A6,A10,A11,Th2,
ZFMISC_1:33;
      end;
    end;
    {.c.} c= C by FINSUB_1:def 5;
    then
A12: {c} c= dom f by FUNCT_2:def 1;
    then
A13: c in dom f by ZFMISC_1:31;
    Im(f,c) c= {e} by A7,RELAT_1:123,XBOOLE_1:7;
    then Im(f,c) = {e} by A12,ZFMISC_1:33;
    then {e} = {f.c} by A13,FUNCT_1:59;
    then f.c = e by ZFMISC_1:3;
    hence thesis by A2,A3,A8,SETWISEO:15;
  end;
A14: X[{}.C];
  for B holds X[B] from SETWISEO:sch 2(A14,A4);
  hence thesis;
end;
