reserve U1,U2,U3 for Universal_Algebra,
  m,n for Nat,
  a for set,
  A for non empty set,
  h for Function of U1,U2;

theorem Th15:
  for U1,U2,h st U1,U2 are_similar for o be OperSymbol of MSSign
  U1,y be Element of Args(o,MSAlg U1) holds (MSAlg h)#y = h * y
proof
  let U1,U2,h;
  assume
A1: U1,U2 are_similar;
  reconsider f1 = (*-->0)*(signature U1) as Function of dom signature(U1), {0}
  * by MSUALG_1:2;
  let o be OperSymbol of MSSign U1;
A2: the carrier' of MSSign U2 = dom signature U2 by MSUALG_1:def 8;
  MSSign U1 = MSSign U2 by A1,Th10;
  then o in dom signature U2 by A2;
  then
A3: o in dom signature U1 by A1;
  then o in dom f1 by FUNCT_2:def 1;
  then
A4: ((*-->0)*(signature U1)).o = (*-->0).((signature U1).o) by FUNCT_1:12;
  let y be Element of Args(o,MSAlg U1);
  set f = MSAlg h;
A5: the carrier of MSSign U1 = {0} by MSUALG_1:def 8;
  set X = dom (h*y);
A6: dom h = the carrier of U1 by FUNCT_2:def 1;
A7: y is FinSequence of the carrier of U1 by Th14;
  then rng y c= the carrier of U1 by FINSEQ_1:def 4;
  then reconsider p = h*y as FinSequence by A7,A6,FINSEQ_1:16;
A8: X = dom y by A7,FINSEQ_3:120;
  the Arity of MSSign U1 = f1 by MSUALG_1:def 8;
  then
A9: the_arity_of o = ((*-->0)*(signature U1)).o by MSUALG_1:def 1;
  (signature U1).o in rng signature U1 by A3,FUNCT_1:def 3;
  then consider n being Element of NAT such that
A10: n = ((signature U1).o);
A11: 0 in {0} by TARSKI:def 1;
A12: now
    0 is Element of {0} by TARSKI:def 1;
    then n |-> 0 is FinSequence of {0} by FINSEQ_2:63;
    then reconsider
l = n |-> 0 as Element of (the carrier of MSSign U1)* by A5,FINSEQ_1:def 11;
    let m;
A13: (the_arity_of o)/.m = l/.m & dom (n |-> 0) = Seg n by A9,A4,A10,
FINSEQ_2:def 6;
    assume m in dom y;
    then m in dom (the_arity_of o) by MSUALG_3:6;
    then
A14: m in dom (n |-> 0) by A9,A4,A10,FINSEQ_2:def 6;
    then l/.m = l.m by PARTFUN1:def 6;
    then (the_arity_of o)/.m = 0 by A14,A13,FUNCOP_1:7;
    hence (f.((the_arity_of o)/.m)) = (( 0.--> h ).0) by A1,Def3,Th10
      .= h by A11,FUNCOP_1:7;
  end;
A15: now
    let m be Nat;
    assume
A16: m in dom y;
    then
A17: m in dom (h*y) by A7,FINSEQ_3:120;
    (f#y).m = (f.((the_arity_of o)/.m)).(y.m) by A16,MSUALG_3:def 6;
    hence (f#y).m = h.(y.m) by A12,A16
      .= p.m by A7,A17,FINSEQ_3:120;
  end;
  dom (f#y) = dom (the_arity_of o) by MSUALG_3:6
    .= dom (n |-> 0) by A9,A4,A10,FINSEQ_2:def 6
    .= Seg n;
  then
A18: f#y is FinSequence by FINSEQ_1:def 2;
A19: dom y = dom (the_arity_of o) by MSUALG_3:6
    .= dom (n |-> 0) by A9,A4,A10,FINSEQ_2:def 6
    .= Seg n;
  dom (f#y) = dom (the_arity_of o) by MSUALG_3:6
    .= dom (n |-> 0) by A9,A4,A10,FINSEQ_2:def 6
    .= X by A7,A19,FINSEQ_3:120;
  hence thesis by A18,A15,A8,FINSEQ_1:13;
end;
