reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));
reserve C for (context of x), C1 for (context of y), C9 for (context of z),
  C11 for (context of x11), C12 for (context of y11), D for context of s,X;
reserve
  S9 for sufficiently_rich non empty non void ManySortedSign,
  s9 for SortSymbol of S9,
  o9 for s9-dependent OperSymbol of S9,
  X9 for non-trivial ManySortedSet of the carrier of S9,
  x9 for (Element of X9.s9);

theorem Th45A:
  for s0 being SortSymbol of S, x0 being Element of X.s0 holds
  the_sort_of t = s & C is x0-omitting & t is x0-omitting implies
  C-sub t is x0-omitting
  proof
    let s0 be SortSymbol of S;
    let x0 be Element of X.s0;
    assume the_sort_of t = s;
    then
A1: C-sub t = (C,[x,s])<-t by SUB;
    assume Z1: Coim(C,[x0,s0]) = {};
    assume Z2: Coim(t,[x0,s0]) = {};
    assume Coim(C-sub t, [x0,s0]) <> {};
    then consider a such that
A2: a in Coim(C-sub t, [x0,s0]) by XBOOLE_0:7;
A3: a in dom(C-sub t) & (C-sub t).a in {[x0,s0]} by A2,FUNCT_1:def 7;
    reconsider a as Element of dom(C-sub t) by A2,FUNCT_1:def 7;
A5: now given q being Node of C, r being Node of t such that
B1:   q in Leaves dom C & C.q = [x,s] & a = q^r;
      (C-sub t).a = t.r by A1,B1,TREES_4:def 7;
      hence contradiction by Z2,A3,FUNCT_1:def 7;
    end;
    per cases by A1,TREES_4:def 7;
    suppose
B3:   a in dom C & C.a <> [x,s];
      then (C-sub t).a = C.a by A1,TREES_4:def 7;
      hence contradiction by Z1,A3,B3,FUNCT_1:def 7;
    end;
    suppose
B2:   a in dom C & C.a = [x,s];
      then reconsider q = a as Node of C;
      reconsider r = {} as Node of t by TREES_1:22;
      q^r = a & q in Leaves dom C by B2,Lem13;
      hence contradiction by A5,B2;
    end;
    suppose ex q being Node of C, r being Node of t st
      q in Leaves dom C & C.q = [x,s] & a = q^r;
      hence contradiction by A5;
    end;
  end;
