
theorem Th3: :: SPpart0:
for X, Y being set, F being a_partition of X, G being a_partition of Y
  st X misses Y holds F \/ G is a_partition of X \/ Y
proof
  let X, Y be set, F be a_partition of X, G be a_partition of Y such that
A1: X misses Y;
   set PR = F \/ G; set XY = X \/ Y;
A2: PR is Subset-Family of XY by Th2;
A3: union PR = union F \/ union G by ZFMISC_1:78
    .= X \/ union G by EQREL_1:def 4 .= X \/ Y by EQREL_1:def 4;
   now
    let A be Subset of XY such that
   A4: A in PR;
     A in F or A in G by A4,XBOOLE_0:def 3;
    hence A <> {};
    let B be Subset of XY such that
   A5: B in PR;
    per cases by A4,A5,XBOOLE_0:def 3;
    suppose A in F & B in F or A in G & B in G;
     hence A = B or A misses B by EQREL_1:def 4;
    end;
    suppose A in F & B in G or A in G & B in F;
     hence A = B or A misses B by A1,XBOOLE_1:64;
    end;
   end;
  hence F \/ G is a_partition of X \/ Y by A2,A3,EQREL_1:def 4;
end;
