reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;

theorem Th97:
  |.z1 + z2.| <= |.z1.| + |.z2.|
proof
A1: 0 <= Sum sqr abs (z1 + z2) by RVSUM_1:86;
A2: 0 <= Sum sqr abs z1 by RVSUM_1:86;
  then
A3: 0 <= sqrt Sum sqr abs z1 by SQUARE_1:def 2;
A4: for k be Nat holds k in Seg n implies 0 <= (mlt(abs z1,abs z2)).k
  proof
    let k be Nat;
    set r = (mlt(abs z1,abs z2)).k;
    assume
A5: k in Seg n;
    then reconsider c1 = z1.k, c2 = z2.k as Element of COMPLEX by Th57;
    (abs z1).k = |.c1.| & (abs z2).k = |.c2.| by A5,Th88;
    then
A6: r = |.c1.|*|.c2.| by RVSUM_1:60;
    0 <= |.c1.| & 0 <= |.c2.| by COMPLEX1:46;
    hence thesis by A6;
  end;
  0 <= (Sum mlt(abs z1,abs z2))^2 by XREAL_1:63;
  then
A7: sqrt(Sum mlt(abs z1,abs z2))^2 <= sqrt((Sum sqr abs z1)*(Sum sqr abs z2
  )) by RVSUM_1:92,SQUARE_1:26;
  len mlt(abs z1,abs z2) = n by CARD_1:def 7;
  then dom mlt(abs z1,abs z2) = Seg n by FINSEQ_1:def 3;
  then Sum mlt(abs z1,abs z2) <= sqrt((Sum sqr abs z1)*(Sum sqr abs z2)) by A4
,A7,RVSUM_1:84,SQUARE_1:22;
  then 2*Sum mlt(abs z1,abs z2) <= 2*sqrt((Sum sqr abs z1)*(Sum sqr abs z2))
  by XREAL_1:64;
  then
  Sum sqr abs z1+(2*Sum mlt(abs z1,abs z2)) <= Sum sqr abs z1+2*sqrt((Sum
  sqr abs z1)*(Sum sqr abs z2)) by XREAL_1:7;
  then
A8: Sum sqr abs z1+(2*Sum mlt(abs z1,abs z2)) + Sum sqr abs z2 <= Sum sqr
  abs z1+2*sqrt((Sum sqr abs z1)*(Sum sqr abs z2)) + Sum sqr abs z2 by
XREAL_1:7;
A9: for k be Nat holds k in Seg n implies (sqr abs (z1 + z2)).k <= (sqr (abs
  z1 + abs z2)).k
  proof
    let k be Nat;
    set r2 = (sqr (abs z1 + abs z2)).k;
    len(abs z1 + abs z2) = n by CARD_1:def 7;
    then
A10: dom (abs z1 + abs z2) = Seg n by FINSEQ_1:def 3;
    assume
A11: k in Seg n;
    then reconsider c12 = (z1 + z2).k as Element of COMPLEX by Th57;
    reconsider abs912 = (abs (z1 + z2)).k as Real;
    0 <= |.c12.| by COMPLEX1:46;
    then
A12: 0 <= abs912 by A11,Th88;
    reconsider abs1 = (abs z1).k, abs2 = (abs z2).k as Real;
    reconsider c1 = z1.k, c2 = z2.k as Element of COMPLEX by A11,Th57;
    reconsider abs12 = (abs z1 + abs z2).k as Real;
    |.c1 + c2.| <= |.c1.| + |.c2.| by COMPLEX1:56;
    then |.c12.| <= |.c1.| + |.c2.| by A11,Th58;
    then |.c12.| <= |.c1.| + abs2 by A11,Th88;
    then |.c12.| <= abs1 + abs2 by A11,Th88;
    then |.c12.| <= abs12 by A11,A10,VALUED_1:def 1;
    then abs912 <= abs12 by A11,Th88;
    then (abs912)^2 <= (abs12)^2 by A12,SQUARE_1:15;
    then (abs912)^2 <= r2 by VALUED_1:11;
    hence thesis by VALUED_1:11;
  end;
A13: 0 <= Sum sqr abs z2 by RVSUM_1:86;
  then
A14: 0 <= sqrt Sum sqr abs z2 by SQUARE_1:def 2;
A15: Sum sqr abs z1 = (sqrt Sum sqr abs z1)^2 by A2,SQUARE_1:def 2;
A16: (sqrt Sum sqr abs z2)^2 = Sum sqr abs z2 by A13,SQUARE_1:def 2;
  Sum sqr (abs z1 + abs z2) = Sum (sqr abs z1 + 2*mlt(abs z1,abs z2) + sqr
  abs z2) by RVSUM_1:68
    .= Sum(sqr abs z1 + 2*mlt(abs z1,abs z2)) + Sum sqr abs z2 by RVSUM_1:89
    .= Sum sqr abs z1 + Sum(2*mlt(abs z1,abs z2)) + Sum sqr abs z2 by
RVSUM_1:89
    .= Sum sqr abs z1 + (2*Sum mlt(abs z1,abs z2))+Sum sqr abs z2 by RVSUM_1:87
;
  then Sum sqr abs (z1 + z2) <= Sum sqr abs z1 + (2*Sum mlt(abs z1,abs z2))+
  Sum sqr abs z2 by A9,RVSUM_1:82;
  then Sum sqr abs (z1 + z2) <= Sum sqr abs z1+2*sqrt((Sum sqr abs z1)*(Sum
  sqr abs z2)) + Sum sqr abs z2 by A8,XXREAL_0:2;
  then Sum sqr abs (z1 + z2) <= Sum sqr abs z1+2*((sqrt Sum sqr abs z1)*(sqrt
  Sum sqr abs z2)) + Sum sqr abs z2 by A2,A13,SQUARE_1:29;
  then sqrt Sum sqr abs (z1 + z2) <= sqrt(((sqrt Sum sqr abs z1) + (sqrt Sum
  sqr abs z2))^2) by A15,A16,A1,SQUARE_1:26;
  hence thesis by A3,A14,SQUARE_1:22;
end;
