reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem
  Z (\/) V = X (\/) Y & X misses Z & Y misses V implies X = V & Y = Z
proof
  assume
A1: Z (\/) V = X (\/) Y;
  assume X misses Z & Y misses V;
  then
A2: X (/\) Z = EmptyMS I & Y (/\) V = EmptyMS I by Th111;
  thus X = X (/\) (Z (\/) V) by Th23,A1,Th14
    .= X (/\) Z (\/) X (/\) V by Th32
    .= (X (\/) Y) (/\) V by A2,Th32
    .= V by A1,Th14,Th23;
  thus Y = Y (/\) (Z (\/) V) by Th23,A1,Th14
    .= Y (/\) Z (\/) Y (/\) V by Th32
    .= (X (\/) Y) (/\) Z by A2,Th32
    .= Z by A1,Th14,Th23;
end;
