reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;

theorem Th35:
  for M being FinSequence of D* holds Values M = union {rng f where
  f is Element of D*: f in rng M}
proof
  let M be FinSequence of D*;
  set R = {rng f where f is Element of D*: f in rng M};
A1: Union rngs M = union rng rngs M by CARD_3:def 4;
  now
    let y be object;
    hereby
      assume y in Values M;
      then consider Y being set such that
A2:   y in Y and
A3:   Y in rng rngs M by A1,TARSKI:def 4;
      consider i being object such that
A4:   i in dom rngs M and
A5:   (rngs M).i = Y by A3,FUNCT_1:def 3;
A6:   i in dom M by A4,FUNCT_6:60;
      then reconsider i as Nat;
      reconsider f = M.i as FinSequence of D by Th34;
A7:   Y = rng f by A5,A6,FUNCT_6:22;
A8:   f in D* by FINSEQ_1:def 11;
      f in rng M by A6,FUNCT_1:def 3;
      then rng f in R by A8;
      hence y in union R by A2,A7,TARSKI:def 4;
    end;
    assume y in union R;
    then consider Y being set such that
A9: y in Y and
A10: Y in R by TARSKI:def 4;
    consider f being Element of D* such that
A11: Y = rng f and
A12: f in rng M by A10;
    consider i being Nat such that
A13: i in dom M & M.i = f by A12,FINSEQ_2:10;
    i in dom rngs M & (rngs M).i = rng f by A13,FUNCT_6:22;
    then rng f in rng rngs M by FUNCT_1:def 3;
    hence y in Values M by A1,A9,A11,TARSKI:def 4;
  end;
  hence thesis by TARSKI:2;
end;
