reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve A1 for Subset of X1,
  A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;
reserve x for Element of [:X1,X2,X3:];

theorem Th54:
  z in [: X1,X2,X3 :] implies ex x1,x2,x3 st x1 in X1 & x2 in X2 &
  x3 in X3 & z = [x1,x2,x3]
proof
  assume z in [: X1,X2,X3 :];
  then z in [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
  then consider x12, x3 being object such that
A1: x12 in [:X1,X2:] and
A2: x3 in X3 and
A3: z = [x12,x3] by ZFMISC_1:def 2;
  consider x1,x2 being object such that
A4: x1 in X1 & x2 in X2 and
A5: x12 = [x1,x2] by A1,ZFMISC_1:def 2;
  z = [x1,x2,x3] by A3,A5;
  hence thesis by A2,A4;
end;
