reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem Th94:
  X c= Y implies [:X,Z:] c= [:Y,Z:] & [:Z,X:] c= [:Z,Y:]
proof
  assume
A1: X c= Y;
  thus [:X,Z:] c= [:Y,Z:]
  proof
    let z;
    assume z in [:X,Z:];
    then consider x,y such that
A2: x in X and
A3: y in Z & z=[x,y] by Def2;
    x in Y by A1,A2;
    hence thesis by A3,Def2;
  end;
  let z;
  assume z in [:Z,X:];
  then consider y,x such that
A4: y in Z and
A5: x in X and
A6: z=[y,x] by Def2;
  x in Y by A1,A5;
  hence thesis by A4,A6,Def2;
end;
