reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem
  for S be locally_directed regular monotone OrderSortedSign, X be
  non-empty ManySortedSet of S, x being set holds x is MinTerm of S,X iff x in
  MinTerms X
proof
  let S be locally_directed regular monotone OrderSortedSign, X be non-empty
  ManySortedSet of S, x be set;
  hereby
    assume x is MinTerm of S,X;
    then reconsider t =x as MinTerm of S,X;
    (PTMin X).t = t by Def32;
    hence x in MinTerms X by FUNCT_2:4;
  end;
  assume x in MinTerms X;
  then consider y being object such that
A1: y in dom (PTMin X) and
A2: x = (PTMin X).y by FUNCT_1:def 3;
  reconsider t = y as Element of TS DTConOSA(X) by A1;
  (PTMin X).t is Element of TS DTConOSA(X);
  then reconsider tx = x as Element of TS DTConOSA(X) by A2;
  (PTMin X).tx = tx by A1,A2,Th43;
  hence thesis by Def32;
end;
