reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem Th10:
  U1 is strict SubAlgebra of U2 & U2 is strict SubAlgebra of U1 implies U1 = U2
proof
  assume that
A1: U1 is strict SubAlgebra of U2 and
A2: U2 is strict SubAlgebra of U1;
  reconsider B = the carrier of U1 as non empty Subset of U2 by A1,Def7;
  the carrier of U2 c= the carrier of U2;
  then reconsider B1 = the carrier of U2 as non empty Subset of U2;
A3: dom Opers(U2,B1) = dom the charact of (U2) by Def6;
A4: for n being Nat st n in dom the charact of(U2) holds (the charact of(U2)
  ).n = (Opers(U2,B1)).n
  proof
    let n be Nat;
    assume
A5: n in dom the charact of(U2);
    then reconsider o =(the charact of(U2)).n as operation of U2 by
FUNCT_1:def 3;
    (Opers(U2,B1)).n = o/.B1 by A3,A5,Def6
      .= o by Th4;
    hence thesis;
  end;
  the carrier of U2 is Subset of U1 by A2,Def7;
  then
A6: B1 = B;
  then the charact of(U1) = Opers(U2,B1) by A1,Def7;
  hence thesis by A1,A2,A6,A3,A4,FINSEQ_1:13;
end;
