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 Th58:
  X1 c= Y1 & X2 c= Y2 & X3 c= Y3 implies [:X1,X2,X3:] c= [:Y1,Y2,
  Y3:]
proof
A1: [:X1,X2,X3:] = [:[:X1,X2:],X3:] & [:Y1,Y2,Y3:] = [:[:Y1,Y2:],Y3:] by
ZFMISC_1:def 3;
  assume X1 c= Y1 & X2 c= Y2;
  then
A2: [:X1,X2:] c= [:Y1,Y2:] by ZFMISC_1:96;
  assume X3 c= Y3;
  hence thesis by A2,A1,ZFMISC_1:96;
end;
