
theorem u1:
for R being Ring
for n,i being Nat st 1 <= i & i <= n holds
i_th_unit_vector(n,R).i = 1.R &
for j being Nat st 1 <= j & j <= n & j <> i holds i_th_unit_vector(n,R).j = 0.R
proof
let R be Ring, n,i be Nat;
set v = i_th_unit_vector(n,R);
assume AS: 1 <= i & i <= n;
A: 1 <= i & i <= len(n |-> 0.R) by AS,FINSEQ_2:132;
D: dom v = Seg n by FINSEQ_2:124;
hence v.i = v/.i by AS,FINSEQ_1:1,PARTFUN1:def 6 .= 1.R by A,FINSEQ_7:8;
now let j be Nat;
assume C: 1 <= j & j <= n & j <> i;
  B: 1 <= j & j <= len(n |-> 0.R) by C,FINSEQ_2:132;
  E: dom(n |-> 0.R) = Seg n by FINSEQ_2:124;
  thus v.j = v/.j by D,C,FINSEQ_1:1,PARTFUN1:def 6
          .= (n |-> 0.R)/.j by C,B,FINSEQ_7:10
          .= (n |-> 0.R).j by E,C,FINSEQ_1:1,PARTFUN1:def 6
          .= (Seg n --> 0.R).j by FINSEQ_2:def 2
          .= 0.R by C,FINSEQ_1:1,FUNCOP_1:7;
  end;
hence thesis;
end;
