reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;

theorem Th43:
  rng f misses rng Col(G,i) & f/.n = G*(m,k) & n in dom f & m in
  dom G implies i<>k
proof
  assume that
A1: rng f /\ rng Col(G,i)={} and
A2: f/.n=G*(m,k) and
A3: n in dom f and
A4: m in dom G and
A5: i=k;
  f.n = G*(m,k) by A2,A3,PARTFUN1:def 6;
  then
A6: G*(m,i) in rng f by A3,A5,FUNCT_1:def 3;
A7: dom G = Seg len G & dom Col(G,i)=Seg len Col(G,i) by FINSEQ_1:def 3;
  len Col(G,i)=len G & Col(G,i).m=G*(m,i) by A4,Def8;
  then G*(m,i) in rng Col(G,i) by A4,A7,FUNCT_1:def 3;
  hence contradiction by A1,A6,XBOOLE_0:def 4;
end;
