
theorem Th20:
for X,Y be non empty set, E1,E2 be Subset of [:X,Y:], p be set holds
  X-section(E1\/E2,p) = X-section(E1,p) \/ X-section(E2,p)
& Y-section(E1\/E2,p) = Y-section(E1,p) \/ Y-section(E2,p)
proof
   let X,Y be non empty set, E1,E2 be Subset of [:X,Y:], p be set;
   now let q be set;
    assume q in X-section(E1\/E2,p); then
    consider y1 be Element of Y such that
A2:  q = y1 & [p,y1] in E1 \/ E2;
    [p,y1] in E1 or [p,y1] in E2 by A2,XBOOLE_0:def 3; then
    q in X-section(E1,p) or q in X-section(E2,p) by A2;
    hence q in X-section(E1,p) \/ X-section(E2,p) by XBOOLE_0:def 3;
   end; then
A3: X-section(E1\/E2,p) c= X-section(E1,p) \/ X-section(E2,p);
   now let q be set;
    assume A4: q in X-section(E1,p) \/ X-section(E2,p);
    per cases by A4,XBOOLE_0:def 3;
    suppose q in X-section(E1,p); then
     consider y1 be Element of Y such that
A5:   q = y1 & [p,y1] in E1;
     [p,y1] in E1 \/ E2 by A5,XBOOLE_0:def 3;
     hence q in X-section(E1\/E2,p) by A5;
    end;
    suppose q in X-section(E2,p); then
     consider y1 be Element of Y such that
A6:   q = y1 & [p,y1] in E2;
     [p,y1] in E1 \/ E2 by A6,XBOOLE_0:def 3;
     hence q in X-section(E1\/E2,p) by A6;
    end;
   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 A3;
   now let q be set;
    assume q in Y-section(E1\/E2,p); then
    consider x1 be Element of X such that
A2:  q = x1 & [x1,p] in E1 \/ E2;
    [x1,p] in E1 or [x1,p] in E2 by A2,XBOOLE_0:def 3; then
    q in Y-section(E1,p) or q in Y-section(E2,p) by A2;
    hence q in Y-section(E1,p) \/ Y-section(E2,p) by XBOOLE_0:def 3;
   end; then
A3: Y-section(E1\/E2,p) c= Y-section(E1,p) \/ Y-section(E2,p);
   now let q be set;
    assume A4: q in Y-section(E1,p) \/ Y-section(E2,p);
    per cases by A4,XBOOLE_0:def 3;
    suppose q in Y-section(E1,p); then
     consider x1 be Element of X such that
A5:   q = x1 & [x1,p] in E1;
     [x1,p] in E1 \/ E2 by A5,XBOOLE_0:def 3;
     hence q in Y-section(E1\/E2,p) by A5;
    end;
    suppose q in Y-section(E2,p); then
     consider x1 be Element of X such that
A6:   q = x1 & [x1,p] in E2;
     [x1,p] in E1 \/ E2 by A6,XBOOLE_0:def 3;
     hence q in Y-section(E1\/E2,p) by A6;
    end;
   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 A3;
end;
