reserve AFV for WeakAffVect;
reserve a,b,c,d,e,f,a9,b9,c9,d9,f9,p,q,r,o,x99 for Element of AFV;
reserve a,b,c for Element of GroupVect(AFV,o);
reserve a,b for Element of GroupVect(AFV,o);
reserve AFV for AffVect,
  o for Element of AFV;

theorem Th50:
  for AFV being strict AffVect holds for o being Element of AFV
  holds AFV = AV(GroupVect(AFV,o))
proof
  let AFV be strict AffVect;
  let o be Element of AFV;
  set X = GroupVect(AFV,o);
  now
    let x,y be object;
    set xy = [x,y];
A1: now
      set V = the carrier of AFV;
      assume
A2:   xy in the CONGR of AFV;
      set VV = [:V,V:];
      xy`2 = y;
      then
A3:   y in VV by A2,MCART_1:10;
      then
A4:   y = [y`1,y`2] by MCART_1:21;
      xy`1 = x;
      then
A5:   x in VV by A2,MCART_1:10;
      then reconsider
      x1 = x`1, x2 = x`2, y1 = y`1, y2 = y`2 as Element of AFV by A3,MCART_1:10
;
      reconsider x19 = x1, x29 = x2, y19 = y1, y29 = y2 as Element of X;
A6:   x = [x`1,x`2] by A5,MCART_1:21;
      then
A7:   x1,x2 // y1,y2 by A2,A4,ANALOAF:def 2;
      x19 # y29 = x29 # y19
      proof
        reconsider z1=x19#y29,z2=x29#y19 as Element of AFV;
        z1 = Padd(o,x1,y2) by Def6;
        then o,x1 // y2,z1 by Def5;
        then x1,o // z1,y2 by Th7;
        then
A8:     o,x2 // y1,z1 by A7,Th12;
        z2 = Padd(o,x2,y1) by Def6;
        hence thesis by A8,Def5;
      end;
      hence [x,y] in CONGRD(X) by A6,A4,TDGROUP:def 2;
    end;
    now
      set V = the carrier of X;
      assume
A9:   xy in CONGRD(X);
      set VV = [:V,V:];
      xy`2 = y;
      then
A10:  y in VV by A9,MCART_1:10;
      then
A11:  y = [y`1,y`2] by MCART_1:21;
      xy`1 = x;
      then
A12:  x in VV by A9,MCART_1:10;
      then reconsider
      x19 = x`1, x29 = x`2, y19 = y`1, y29 = y`2 as Element of X by A10,
MCART_1:10;
      set z19 = x19 # y29, z29 = x29 # y19;
      reconsider x1 = x19, x2 = x29, y1 = y19, y2 = y29 as Element of AFV;
      reconsider z1=z19,z2=z29 as Element of AFV;
A13:  z2 = Padd(o,x2,y1) by Def6;
      z1 = Padd(o,x1,y2) by Def6;
      then
A14:  o,x1 // y2,z1 by Def5;
A15:  x = [x`1,x`2] by A12,MCART_1:21;
      then z19=z29 by A9,A11,TDGROUP:def 2;
      then o,x2 // y1,z1 by A13,Def5;
      then x1,x2 // y1,y2 by A14,Th12;
      hence xy in the CONGR of AFV by A15,A11,ANALOAF:def 2;
    end;
    hence [x,y] in CONGRD(X) iff [x,y] in the CONGR of AFV by A1;
  end;
  then the carrier of AV(X) = the carrier of AFV & CONGRD(X) = the CONGR of
  AFV by RELAT_1:def 2,TDGROUP:4;
  hence thesis by TDGROUP:4;
end;
