reserve i,j for Nat;
reserve i,j for Nat,
  x for variable,
  l for quasi-loci;
reserve C for initialized ConstructorSignature,
  c for constructor OperSymbol of C;

theorem
  for e being constructor expression of C holds main-constr e in constrs e
  proof
    let e be constructor expression of C;
A1: main-constr e = (e.{})`1 by Def9;
    {} in dom e by TREES_1:22; then
    e.{} in rng e by FUNCT_1:def 3; then
A2: main-constr e in proj1 rng e by A1,MCART_1:86;
    main-constr e is constructor OperSymbol of C by Def11; then
    main-constr e in the set of all c;
    hence main-constr e in constrs e by A2,XBOOLE_0:def 4;
  end;
