
theorem Th4: :: U5:
for A, B, C, D, E, x being set holds
 x in A \/ B \/ C \/ D \/ E iff x in A or x in B or x in C or x in D or x in E
proof
 let A, B, C, D, E, x be set;
 hereby
  assume x in A \/ B \/ C \/ D \/ E;
  then x in A \/ B \/ C \/ D or x in E by XBOOLE_0:def 3;
  then x in A \/ B \/ C or x in D or x in E by XBOOLE_0:def 3;
  then x in A \/ B or x in C or x in D or x in E by XBOOLE_0:def 3;
  hence x in A or x in B or x in C or x in D or x in E by XBOOLE_0:def 3;
 end;
 assume A1: x in A or x in B or x in C or x in D or x in E;
 per cases by A1;
 suppose x in A;
   then x in A \/ B by XBOOLE_0:def 3;
   then x in A \/ B \/ C by XBOOLE_0:def 3;
   then x in A \/ B \/ C \/ D by XBOOLE_0:def 3;
  hence x in A \/ B \/ C \/ D \/ E by XBOOLE_0:def 3;
 end;
 suppose x in B;
   then x in A \/ B by XBOOLE_0:def 3;
   then x in A \/ B \/ C by XBOOLE_0:def 3;
   then x in A \/ B \/ C \/ D by XBOOLE_0:def 3;
  hence x in A \/ B \/ C \/ D \/ E by XBOOLE_0:def 3;
 end;
 suppose x in C;
   then x in A \/ B \/ C by XBOOLE_0:def 3;
   then x in A \/ B \/ C \/ D by XBOOLE_0:def 3;
  hence x in A \/ B \/ C \/ D \/ E by XBOOLE_0:def 3;
 end;
 suppose x in D;
   then x in A \/ B \/ C \/ D by XBOOLE_0:def 3;
  hence x in A \/ B \/ C \/ D \/ E by XBOOLE_0:def 3;
 end;
 suppose x in E;
  hence x in A \/ B \/ C \/ D \/ E by XBOOLE_0:def 3;
 end;
end;
