
theorem Th43:
for X1,X2 be non empty set, A,B be Subset of [:X1,X2:], p be set holds
  X-section(A \ B,p) = X-section(A,p) \ X-section(B,p)
& Y-section(A \ B,p) = Y-section(A,p) \ Y-section(B,p)
proof
   let X1,X2 be non empty set, E1,E2 be Subset of [:X1,X2:], p be set;
   now let q be set;
    assume q in X-section(E1 \ E2,p); then
    q in {y where y is Element of X2: [p,y] in E1 \ E2} by MEASUR11:def 4; then
A1: ex y be Element of X2 st q = y & [p,y] in E1 \ E2; then
    [p,q] in E1 & not [p,q] in E2 by XBOOLE_0:def 5; then
    q in {y where y is Element of X2: [p,y] in E1} by A1; then
A3: q in X-section(E1,p) by MEASUR11:def 4;
    now assume q in X-section(E2,p); then
     q in {y where y is Element of X2: [p,y] in E2} by MEASUR11:def 4; then
     ex y be Element of X2 st q = y & [p,y] in E2;
     hence contradiction by A1,XBOOLE_0:def 5;
    end;
    hence q in X-section(E1,p) \ X-section(E2,p) by A3,XBOOLE_0:def 5;
   end; then
A4:X-section(E1 \ E2,p) c= X-section(E1,p) \ X-section(E2,p);

   now let q be set;
    assume q in X-section(E1,p) \ X-section(E2,p); then
    q in X-section(E1,p) & not q in X-section(E2,p) by XBOOLE_0:def 5; then
A5: q in {y where y is Element of X2: [p,y] in E1} &
    not q in {y where y is Element of X2: [p,y] in E2} by MEASUR11:def 4; then
A6: ex y be Element of X2 st q = y & [p,y] in E1; then
    not [p,q] in E2 by A5; then
    [p,q] in E1 \ E2 by A6,XBOOLE_0:def 5; then
    q in {y where y is Element of X2: [p,y] in E1 \ E2} by A6;
    hence q in X-section(E1 \ E2,p) by MEASUR11:def 4;
   end; then
   X-section(E1,p) \ X-section(E2,p) c= X-section(E1 \ E2,p);
   hence X-section(E1 \ E2,p) = X-section(E1,p) \ X-section(E2,p) by A4;

   now let q be set;
    assume q in Y-section(E1 \ E2,p); then
    q in {x where x is Element of X1: [x,p] in E1 \ E2} by MEASUR11:def 5; then
B1: ex x be Element of X1 st q = x & [x,p] in E1 \ E2; then
    [q,p] in E1 & not [q,p] in E2 by XBOOLE_0:def 5; then
    q in {x where x is Element of X1: [x,p] in E1} by B1; then
B3: q in Y-section(E1,p) by MEASUR11:def 5;
    now assume q in Y-section(E2,p); then
     q in {x where x is Element of X1: [x,p] in E2} by MEASUR11:def 5; then
     ex x be Element of X1 st q = x & [x,p] in E2;
     hence contradiction by B1,XBOOLE_0:def 5;
    end;
    hence q in Y-section(E1,p) \ Y-section(E2,p) by B3,XBOOLE_0:def 5;
   end; then
B4:Y-section(E1 \ E2,p) c= Y-section(E1,p) \ Y-section(E2,p);

   now let q be set;
    assume q in Y-section(E1,p) \ Y-section(E2,p); then
    q in Y-section(E1,p) & not q in Y-section(E2,p) by XBOOLE_0:def 5; then
B5: q in {x where x is Element of X1: [x,p] in E1} &
    not q in {x where x is Element of X1: [x,p] in E2} by MEASUR11:def 5; then
B6: ex x be Element of X1 st q = x & [x,p] in E1; then
    not [q,p] in E2 by B5; then
    [q,p] in E1 \ E2 by B6,XBOOLE_0:def 5; then
    q in {x where x is Element of X1: [x,p] in E1 \ E2} by B6;
    hence q in Y-section(E1 \ E2,p) by MEASUR11:def 5;
   end; then
   Y-section(E1,p) \ Y-section(E2,p) c= Y-section(E1 \ E2,p);
   hence Y-section(E1 \ E2,p) = Y-section(E1,p) \ Y-section(E2,p) by B4;
end;
