reserve n,i,j,k,l for Nat;
reserve D for non empty set;
reserve c,d for Element of D;
reserve p,q,q9,r for FinSequence of D;
reserve RAS for MidSp-like non empty ReperAlgebraStr over n+2;
reserve a,b,d,pii,p9i for Point of RAS;
reserve p,q for Tuple of (n+1),RAS;
reserve m for Nat of n;
reserve W for ATLAS of RAS;
reserve v for Vector of W;
reserve x,y for Tuple of (n+1),W;
reserve RAS for ReperAlgebra of n;
reserve a,b,pm,p9m,p99m for Point of RAS;
reserve p for Tuple of (n+1),RAS;
reserve W for ATLAS of RAS;
reserve v for Vector of W;
reserve x for Tuple of (n+1),W;

theorem Th32:
  (a,x).W = p & m<=n implies (a,(x+*(m+1,x.m))).W = (p+*(m+1,p.m))
proof
  assume that
A1: (a,x).W = p and
A2: m<=n;
  W.(a,p) = x by A1,Th15;
  then W.(a,(p+*(m+1,p.m))) = (x+*(m+1,x.m)) by A2,Th31;
  hence thesis by Th15;
end;
