reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

theorem Th26:
  V is Submodule of X & X is Submodule of Y implies V is Submodule of Y
proof
  assume that
A1: V is Submodule of X and
A2: X is Submodule of Y;
  the carrier of V c= the carrier of X & the carrier of X c= the carrier
  of Y by A1,A2,Def2;
  then
A3: the carrier of V c= the carrier of Y;
A4: the addF of V = (the addF of Y) |([:the carrier of V, the carrier of V:]
  qua set)
  proof
    set AY = the addF of Y;
    set VX = the carrier of X;
    set AX = the addF of X;
    set VV = the carrier of V;
    set AV = the addF of V;
    VV c= VX by A1,Def2;
    then
A5: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
    AV = AX||VV by A1,Def2;
    then AV = AY||VX||VV by A2,Def2;
    hence thesis by A5,FUNCT_1:51;
  end;
  set MY = the rmult of Y;
  set MX = the rmult of X;
  set MV = the rmult of V;
  set VX = the carrier of X;
  set VV = the carrier of V;
  VV c= VX by A1,Def2;
  then
A6: [:VV,the carrier of R:] c= [:the carrier of X,the carrier of R:] by
ZFMISC_1:95;
  MV = MX |([:VV,the carrier of R:] qua set) by A1,Def2;
  then MV = (MY |([:VX,the carrier of R:] qua set)) |([:VV,the carrier of R:]
  qua set) by A2,Def2;
  then
A7: MV = MY |([:VV,the carrier of R:] qua set) by A6,FUNCT_1:51;
  0.V = 0.X by A1,Def2;
  then 0.V = 0.Y by A2,Def2;
  hence thesis by A3,A4,A7,Def2;
end;
