
theorem Th25:
for X,Y be non empty set, A be Element of bool [:X,Y:], x,y be set
 st x in X & y in Y holds
  ([x,y] in A iff x in Y-section(A,y)) & ([x,y] in A iff y in X-section(A,x))
proof
   let X,Y be non empty set, E be Element of bool [:X,Y:], x,y be set;
   assume A1: x in X & y in Y;
A2:now assume y in X-section(E,x); then
    y in {y where y is Element of Y: [x,y] in E} by MEASUR11:def 4; then
    ex y1 be Element of Y st y = y1 & [x,y1] in E;
    hence [x,y] in E;
   end;
A3:now assume [x,y] in E; then
    y in {y where y is Element of Y: [x,y] in E} by A1;
    hence y in X-section(E,x) by MEASUR11:def 4;
   end;
A4:now assume x in Y-section(E,y); then
    x in {x where x is Element of X: [x,y] in E} by MEASUR11:def 5; then
    ex x1 be Element of X st x = x1 & [x1,y] in E;
    hence [x,y] in E;
   end;
   now assume [x,y] in E; then
    x in {x where x is Element of X: [x,y] in E} by A1;
    hence x in Y-section(E,y) by MEASUR11:def 5;
   end;
   hence thesis by A2,A3,A4;
end;
