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Datei: product_sections.pvs   Sprache: PVS

%------------------------------------------------------------------------------
% Sections of sets
%
%     Author: David Lester, Manchester University, NIA, Université Perpignan
%
% All references are to SK Berberian "Fundamentals of Real Analysis",
% Springer, 1991
%
%     Version 1.0            1/5/07   Initial Version
%------------------------------------------------------------------------------

product_sections[T1,T2:TYPE]: THEORY

BEGIN

  X,Y: VAR set[[T1,T2]]
  a:   VAR T1
  b:   VAR T2

  IMPORTING topology@cross_product[T1,T2]

  x_section_emptyset:  LEMMA x_section(emptyset,a) = emptyset
  x_section_complement: LEMMA
    x_section(complement(X),a) = complement(x_section(X,a))
  x_section_union: LEMMA
    x_section(union(X,Y),a) = union(x_section(X,a),x_section(Y,a))
  x_section_intersection: LEMMA
    x_section(intersection(X,Y),a)
      = intersection(x_section(X,a),x_section(Y,a))
  x_section_disjoint: LEMMA
    disjoint?(X,Y) => disjoint?(x_section(X,a),x_section(Y,a))

  y_section_emptyset:  LEMMA y_section(emptyset,b) = emptyset
  y_section_complement: LEMMA
    y_section(complement(X),b) = complement(y_section(X,b))
  y_section_union: LEMMA
    y_section(union(X,Y),b) = union(y_section(X,b),y_section(Y,b))
  y_section_intersection: LEMMA
    y_section(intersection(X,Y),b)
      = intersection(y_section(X,b),y_section(Y,b))
  y_section_disjoint: LEMMA
    disjoint?(X,Y) => disjoint?(y_section(X,b),y_section(Y,b))

END product_sections

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