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 Th11:
  for U1,U2 be SubAlgebra of U0 st the carrier of U1 c= the
  carrier of U2 holds U1 is SubAlgebra of U2
proof
  let U1,U2 be SubAlgebra of U0;
A1: dom the charact of(U1) = dom the charact of(U0) by Th7;
  reconsider B1 = the carrier of U1 as non empty Subset of U0 by Def7;
  assume
A2: the carrier of U1 c= the carrier of U2;
  hence the carrier of U1 is Subset of U2;
  let B be non empty Subset of U2;
  assume
A3: B = the carrier of U1;
  reconsider B2 = the carrier of U2 as non empty Subset of U0 by Def7;
A4: the charact of(U2) = Opers(U0,B2) by Def7;
A5: B2 is opers_closed by Def7;
A6: dom Opers(U2,B) = dom the charact of(U2) by Def6;
A7: dom the charact of(U2)= dom the charact of(U0) by Th7;
A8: B1 is opers_closed by Def7;
A9: B is opers_closed
  proof
    let o2 be operation of U2;
    let s be FinSequence of B;
    consider n being Nat such that
A10: n in dom the charact of(U2) and
A11: (the charact of(U2)).n = o2 by FINSEQ_2:10;
    reconsider o0 = (the charact of(U0)).n as operation of U0 by A7,A10,
FUNCT_1:def 3;
A12: arity o2 = arity o0 by A10,A11,Th6;
    rng s c= B by FINSEQ_1:def 4;
    then
 rng s c= B2 by XBOOLE_1:1;
    then s is FinSequence of B2 by FINSEQ_1:def 4;
    then reconsider s2 = s as Element of B2* by FINSEQ_1:def 11;
    reconsider s1 = s as Element of B1* by A3,FINSEQ_1:def 11;
    assume
A13: arity o2 = len s;
    set tb2 = (arity o0)-tuples_on B2;
A14: B2 is_closed_on o0 by A5;
A15: o2 = o0/.B2 by A4,A10,A11,Def6
      .= o0 | tb2 by A14,Def5;
A16: B1 is_closed_on o0 by A8;
    tb2 = {w where w is Element of B2*: len w = arity o0} by FINSEQ_2:def 4;
    then s2 in tb2 by A13,A12;
    then o2.s = o0.s1 by A15,FUNCT_1:49;
    hence thesis by A3,A13,A16,A12;
  end;
A17: the charact of(U1) = Opers(U0,B1) by Def7;
  now
    let n be Nat;
    assume
A18: n in dom the charact of(U0);
    then reconsider o0 = (the charact of(U0)).n as operation of U0 by
FUNCT_1:def 3;
    reconsider o2 = (the charact of(U2)).n as operation of U2 by A7,A18,
FUNCT_1:def 3;
A19: o2 = o0/.B2 & arity o2 = arity o0 by A4,A7,A18,Def6,Th6;
A20: B is_closed_on o2 by A9;
A21: B2 is_closed_on o0 by A5;
A22: B1 is_closed_on o0 by A8;
    thus (the charact of(U1)).n = o0/.B1 by A17,A1,A18,Def6
      .= o0 | (arity o0)-tuples_on B1 by A22,Def5
      .= o0 | (((arity o0)-tuples_on B2) /\ ((arity o0)-tuples_on B1)) by A2,
MARGREL1:21
      .= (o0 | (arity o0)-tuples_on B2) | ((arity o0)-tuples_on B) by A3,
RELAT_1:71
      .= (o0 /. B2) | ((arity o0)-tuples_on B) by A21,Def5
      .= o2 /. B by A20,A19,Def5
      .= Opers(U2,B).n by A7,A6,A18,Def6;
  end;
  hence the charact of(U1) = Opers(U2,B) by A1,A7,A6,FINSEQ_1:13;
  thus thesis by A9;
end;
