reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem
  sqr cosec x_r-seq(m) = 1 + sqr cot x_r-seq(m)
  proof
    set f = x_r-seq(m);
A1: len sqr cosec f = len cosec f by CARD_1:def 7;
A2: len cosec f = len f by CARD_1:def 7;
A3: len f = m by Th19;
A4: len(1 + sqr cot x_r-seq(m)) = len sqr cot x_r-seq(m) by CARD_1:def 7;
A5: len sqr cot x_r-seq(m) = len cot x_r-seq(m) by CARD_1:def 7;
A6: len cot x_r-seq(m) = len x_r-seq(m) by CARD_1:def 7;
    thus len sqr cosec f = len(1 + sqr cot x_r-seq(m))
    by A1,A2,A4,A5,CARD_1:def 7;
    let k such that
A7: 1 <= k and
A8: k <= len sqr cosec f;
A9: k in dom f by A1,A2,A7,A8,FINSEQ_3:25;
    then
A10: (cosec f).k = cosec(f.k) by Def4;
A11: (sqr cot f).k = ((cot f).k)^2 by VALUED_1:11;
A12: (cot f).k = cot(f.k) by A9,Def3;
A13: 0 < f.k by A1,A2,A3,A7,A8,Th21;
A14: f.k < PI/2 by A1,A2,A3,A7,A8,Th21;
    PI/2 < PI/1 by XREAL_1:76;
    then f.k < PI by A14,XXREAL_0:2;
    then f.k in ].0,PI.[ by A13,XXREAL_1:4;
    then sin(f.k) <> 0 by COMPTRIG:7;
    then
A15: (cosec(f.k))^2 = 1+(cot(f.k))^2 by SIN_COS5:14;
    k in dom (1 + sqr cot f) by A1,A2,A4,A5,A6,A7,A8,FINSEQ_3:25;
    hence (1 + sqr cot f).k = ((cosec f).k)^2
    by A10,A11,A12,A15,VALUED_1:def 2
    .= (sqr cosec f).k by VALUED_1:11;
  end;
