reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th37:
  for A being included_in_Seg set holds len Sgm A = card A
  proof
    let A be included_in_Seg set;
A1: dom Sgm A = Seg len Sgm A by FINSEQ_1:def 3;
A2: card Seg len Sgm A = len Sgm A by FINSEQ_1:57;
A4: rng Sgm A = A by FINSEQ_1:def 14;
    Seg len Sgm A,A are_equipotent by A1,A4;
    hence thesis by A2,CARD_1:5;
  end;
